tag:blogger.com,1999:blog-67927106717334455932018-03-02T09:21:53.938-08:00Summer of GödelDavid Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.comBlogger23125tag:blogger.com,1999:blog-6792710671733445593.post-21172722278909262072014-12-31T12:34:00.000-08:002014-12-31T12:34:10.232-08:00Almost all integers contain the digit 3This is my last chance to blog this year. However, I did not get back into the proof of Gödel's incompleteness theorem yet. So instead, this post is motivated by this&nbsp;<a href="https://www.youtube.com/watch?v=UfEiJJGv4CE" target="_blank">Numberphile video</a>,&nbsp;which explains why almost all integers contain the digit 3. More precisely, the percentage of integers that contain at least one occurrence of the digit '3' in the set \(\mathbb{N}_u = \{0,1,2,3,\cdots,u-1\}\) goes to 100% as the upper bound \(u\) goes to infinity.<br /><br />We are going to attack this problem with two different approaches. First, we'll use a recurrence relation approach. Second, we'll go over the combinatorial proof discussed in the video. Finally, we'll check that the two approaches do give the same answer.<br /><br />In both approaches, we will use values of \(u\) that are powers of 10. So if \(u = 10^n\), for \(n \in \mathbb{N}^+\), then the set under consideration \(\mathbb{N}_{10^n}\), which we denote \(A_n\), is the set of all of the (non-negative) integers containing at most \(n\) digits.<br /><br /><b>Recurrence relation approach</b><br /><br />In this approach, we use \(S_n\), for \(n \in \mathbb{N}^+\),&nbsp;to denote the subset of \(A_n\) of all of the integers that contain at least one occurrence of the digit '3' and we use \(T_n\) to denote \(|S_n|\), the cardinality of \(S_n\). <br /><ul><li>Since \(A_1=\{0,1,2,3,4,5,6,7,8,9\}\), \(S_1=\{3\}\) and \(T_1=1\).</li><li>Since \(A_2=\{0,1,2,3,4,\cdots,98,99\}\), \(S_2=\{ 3,13,23,30,31,32,33,34,\cdots,39,43,53,63,73,83,93\}\)\(=\{ 3,13,23,43,53,63,73,83,93\}\)\( \cup \{30,31,32,33,34,\cdots,39\}\). In other words, we can split the set \(S_2\) into two subsets, one whose elements all start with the digit '3' &nbsp;(there are exactly 10 such integers), and the other one whose elements do not start with the digit '3' (there are exactly \(9\cdot T_1\) of those, since if the first digit is not '3' then the least significant digit must belong to \(S_1\)). Thus, \(T_2 = 10 + 9\cdot T_1 = 10 + 9 = 19\).</li><li>Similarly, since \(A_3=\{0,1,2,3,4,\cdots,998,999\}\), \(S_3\) is the union of two sets, &nbsp;one whose elements all start with the digit '3' &nbsp;(there are exactly \(10^2\) such integers), and the other one whose elements do not start with the digit '3' (there are exactly \(9\cdot T_2\) of those, since if the most significant digit is not '3' then the number formed by the other digits must belong to \(S_2\)).&nbsp;Thus, \(T_3 = 10^2 + 9\cdot T_2= 100 + 9\cdot 19 = 271\).</li><li>Since this reasoning clearly generalizes to all values of \(n \in \mathbb{N}^+\), we obtain the following recurrence relation:</li></ul>\[ \left\{ \begin{array}{ll}<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; T_1 = 1 &amp; \\<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; T_n = 10^{n-1} + 9T_{n-1} &amp; \text{if}\ n&gt;1<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\end{array} \right. &nbsp; &nbsp; &nbsp; <br />\] Now, let's solve this recurrence relation by iteration to get a closed-form formula for \(T_n\):<br /><br />\(<br />\begin{array}{l@{0pt}l}<br />T_1 &nbsp;&amp; = 1\\<br />T_2 = 10^1 + 9T_1 &nbsp;&amp; = 10+9 \\<br />T_3 = 10^2 + 9T_2 &nbsp;&amp; = 10^2+9(10+9) \\<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = 10^2 + 9\cdot 10 + 9^2 \\<br />T_4 = 10^3 + 9T_3 &nbsp;&amp; = 10^3+9(10^2 + 9\cdot 10 + 9^2) \\<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = 10^3+9\cdot 10^2 + 9^2 \cdot 10 + 9^3\\<br />T_5 = 10^4 + 9T_4 &nbsp;&amp; = 10^4+9(10^3+9\cdot 10^2 + 9^2 \cdot 10 + 9^3)\\<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = 10^4+9\cdot 10^3+9^2\cdot 10^2 + 9^3 \cdot 10 + 9^4\\<br />\cdots &amp;\\<br />T_n = 10^{n-1} + 9T_{n-1} &amp; = 10^{n-1}+9\cdot 10^{n-2}+9^2\cdot 10^{n-3} + \cdots + 9^{n-2}\cdot 10^1 + 9^{n-1}\\<br />\end{array}<br />\)<br /><br />In conclusion, \(\displaystyle T_n = \sum_{i=0}^{n-1} \left( &nbsp;9^{n-i-1}\cdot 10^i\right) \) for \(n \in \mathbb{N}^+\)<br /><br />Since this formula is rather ugly, let's turn to the combinatorial approach discussed in the&nbsp;<a href="https://www.youtube.com/watch?v=UfEiJJGv4CE" target="_blank">Numberphile video</a>.<br /><br /><b>Combinatorial approach</b><br /><div><b><br /></b></div>It is easy to determine the total number of integers with exactly \(n\) digits without having to enumerate them, namely \(10 \times 10 \times \cdots \times 10 = 10^n\), &nbsp;since there are exactly 10 digits to choose from for each position in the \(n\)-digit number. Note that this count actually includes all of the numbers with leading zeros, such as 01 and 00023, which are identical to 1 and 23, respectively. In other words, what we really have is &nbsp;\(\left|A_n\right|= 10^n\).<br /><br />Similarly, we can compute the total number of integers with at most \(n\) digits that do not contain the digit '3', namely \(9 \times 9 \times \cdots \times 9 = 9^n\), since there are now only 9 digits to choose from at each position in the integer.<br /><br />In conclusion, \(T_n = 10^n -9^n\).<br /><br />This is both a much nicer and easier-to-derive closed-form formula than the one we obtained with the recurrence relation approach. But are the two formulas equal?<br /><br /><b>Let's check our work</b><br /><br />Let \(P(n) \), for \(n \in \mathbb{N}^+\), denote: \(\displaystyle \sum_{i=0}^{n-1} \left( &nbsp;9^{n-i-1}\cdot 10^i\right) = 10^n - 9^n\)<br /><br />We now prove by mathematical induction that \( P(n)\) holds for &nbsp;\(n \in \mathbb{N}^+.\)<br /><br />Basis: <br /><ul><li>&nbsp;\(\displaystyle \sum_{i=0}^{1-1} \left( &nbsp;9^{1-i-1}\cdot 10^i\right) = &nbsp;\sum_{i=0}^{0} \left( &nbsp;9^{1-i-1}\cdot 10^i\right) = 9^{1-0-1}\cdot 10^0 = 9^0 \cdot 10^0 =1 \)</li><li>\( 10^1 - 9^1 = 10 - 9 = 1\)</li><li>Therefore \(P(1)\) holds</li></ul>Inductive step: <br /><ul><li>Assume that \(P(n)\) holds&nbsp;for any \(n \in \mathbb{N}^+\), that is: \[ \sum_{i=0}^{n-1}&nbsp;\left( &nbsp;9^{n-i-1}\cdot 10^i\right) = 10^n - 9^n \quad \text{[inductive hypothesis]} \] </li><li>Let's compute the left-hand side of \(P(n+1)\): <div>\[<br />\begin{array}{l@{0pt}l}<br />\sum_{i=0}^{n} \left( &nbsp;9^{n-i}\cdot 10^i\right) &amp; = 9\left(\frac{1}{9}\sum_{i=0}^{n} \left( &nbsp;9^{n-i}\cdot 10^i\right)\right)\\<br />&nbsp; &nbsp; &amp; = 9\left(\sum_{i=0}^{n} \left( &nbsp;9^{n-i-1}\cdot 10^i\right)\right)\\<br />&nbsp; &nbsp;&amp; = 9\left(\sum_{i=0}^{n-1} \left( &nbsp;9^{n-i-1}\cdot 10^i\right)&nbsp;&nbsp;+&nbsp;\sum_{i=n}^{n} \left( &nbsp;9^{n-i-1}\cdot 10^i\right)&nbsp;\right)\\<br />&nbsp; &nbsp;&amp; = 9\left(\sum_{i=0}^{n-1} \left( &nbsp;9^{n-i-1}\cdot 10^i\right)&nbsp;&nbsp;+ \left( &nbsp;9^{n-n-1}\cdot 10^n\right)&nbsp;\right)\\<br />&nbsp;&amp; = 9\left(\sum_{i=0}^{n-1} \left( &nbsp;9^{n-i-1}\cdot 10^i\right)&nbsp;&nbsp;+ \frac{10^n}{9} \right)\\<br />&amp; &nbsp;= 9\left( \left(10^n - 9^n\right)&nbsp;&nbsp;+ \frac{10^n}{9} \right) \quad \text{by the inductive hypothesis} \\<br />&nbsp;&amp; = 9\cdot 10^n - 9\cdot 9^n + 10^n\\<br />&nbsp;&amp; = 10\cdot 10^n - 9\cdot 9^n\\<br />&nbsp;&amp; = 10^{n+1} - 9^{n+1}\\<br />\end{array}<br />\]</div></li><li>Therefore \(P(n+1)\) holds.</li></ul><b><br /></b><b>Main result</b><br /><br />We are now in a position to prove our main result, namely that the percentage of integers that contain the digit '3' in the set \(A_n\) &nbsp;goes to 100% as \(n\) goes to infinity. This percentage is equal to<br />\(100\times\frac{T_n}{\left|A_n\right|}=100\times\left(\frac{10^n-9^n}{10^n}\right)=100\times\left(1-\frac{9^n}{10^n}\right)= 100 - 100\left(\frac{9}{10}\right)^n\).<br />And this percentage goes to 100% as \(n\) goes to infinity, since \(\displaystyle \lim_{n \to \infty}\left( \frac{9}{10}\right)^n = 0\).<br /><br /><b>Discussion</b><br /><b><br /></b>Of course, all of the proofs above still hold if we had picked the digit '8' (say) instead of the digit '3'. In other words, it is also true that almost all integers contain the digit '8'. So, if we use \(E_n\) to denote the cardinality of the subset of \(A_n\) of all of the integers that contain at least one occurrence of the digit '8', \(E_n = 10^n-9^n\). Similarly for the digit '5' (say): \(F_n = 10^n-9^n\).<br /><br />But, if both \(100\times\frac{T_n}{\left|A_n\right|}\) and &nbsp;\(100\times\frac{E_n}{\left|A_n\right|}\) go to 100% as \(n\) goes to infinity, the sum of these two percentages will eventually exceed 100%! That is true. However, this sum double counts all of the integers that contain both the digit '3' and the digit '8'. The correct percentage of such integers is \(100\times\frac{10^n-8^n}{10^n}\), which also goes to 100% as \(n\) goes to infinity.<br /><br />So there is no paradox here. When \(n\) gets large enough, almost all of the integers will contain all of the digits 0 through 9. Note that all of these results are asymptotic. Indeed, it takes relatively large values of \(n\) for these percentages to get anywhere close to 100%. For example, for the percentage of integers that contain the digit '3' to reach 90%, 95%, 99% and 99.99%, the number \(n\) of digits in the integer must be larger than 21, 28, 43 and 87, respectively.<img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/nhFtPEU91Z8" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2014/12/almost-all-integers-contain-digit-3.htmltag:blogger.com,1999:blog-6792710671733445593.post-9642938294997786282013-07-09T12:26:00.001-07:002013-07-09T12:26:36.528-07:00Robinson Arithmetic is Σ1-completeRecall that <a href="http://summerofgodel.blogspot.com/2013/06/q-robinson-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Q (i.e., Robinson Arithmetic)</span></a> is an <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html" target="_blank"><span class="Apple-style-span" style="color: blue;">axiomatized formal theory (AFT)</span> </a>of arithmetic couched in <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;">the interpreted formal language <i>L<sub>A</sub></i></span></a>. Let L be a subset of <i>L<sub>A</sub></i> and let T be some AFT of arithmetic.<br /><br />We say that T is <b>L-sound</b> iff, for any sentence φ in L, if&nbsp;T ⊢ φ, then&nbsp;φ is true.<br /><br />We say that T is&nbsp;<b>L-complete</b>&nbsp;iff, for any sentence φ in L, if φ is true, then T ⊢ φ.<br /><br />In chapter 9, <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372464710&amp;sr=1-2" target="_blank"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> defines a subset of&nbsp;<i>L<sub>A</sub></i>, called Σ<sub>1</sub>. Then,&nbsp;using the fact that&nbsp;<a href="http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-is-order-adequate.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Q is order-adequate</span></a>, he proves that&nbsp;Q is Σ<sub>1</sub>-complete. This is important because the well-formed formulas (<b>wff</b>'s) of&nbsp;Σ<sub>1&nbsp;</sub>can express the decidable numerical properties and relations, and therefore Q will be&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/consistent-sufficiently-strong-formal.html" target="_blank"><span class="Apple-style-span" style="color: blue;">sufficiently strong</span></a>. Now to the details...<br /><br />First, let's define a few interesting subsets of&nbsp;<i>L<sub>A</sub></i>:<br /><a name='more'></a><ol><li>The set of <b>atomic Δ<sub>0</sub> wff's</b> is the set of wff's of the form σ = τ or&nbsp;σ ≤ τ, where σ and τ are terms of&nbsp;<i>L<sub>A</sub></i>.</li><li>The set of&nbsp;<b>Δ<sub>0</sub>&nbsp;wff's</b>&nbsp;is defined inductively by the following rules:</li><ul><li>Every atomic&nbsp;Δ<sub>0</sub>&nbsp;wff&nbsp;is&nbsp;a Δ<sub>0</sub>&nbsp;wff.</li><li>If α and β are Δ<sub>0</sub>&nbsp;wff's, then so are ¬α, α ∧ β,&nbsp;α&nbsp;∨&nbsp;β, and&nbsp;α&nbsp;→&nbsp;β.</li><li>If α is a Δ<sub>0</sub>&nbsp;wff, then so&nbsp;are (∀χ ≤ κ) α and&nbsp;&nbsp;(∃χ ≤ κ) α, where&nbsp;χ is any variable that appears free in&nbsp;α and&nbsp;κ is a numeral or a variable that is different from&nbsp;χ (see <a href="http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-captures-less-than.html" target="_blank"><span class="Apple-style-span" style="color: blue;">this post</span></a> for the definition of these bounded quantifiers).</li><li>Nothing else is a&nbsp;Δ<sub>0</sub>&nbsp;wff.</li></ul><li>A wff is <b>strictly&nbsp;Σ<sub>1</sub></b>&nbsp;iff it is of the form&nbsp;∃α∃β...∃ω φ, where&nbsp;φ is Δ<sub>0</sub>&nbsp;and&nbsp;α, β, ... , ω are one or more distinct variables that appear free in&nbsp;φ.</li><li>A wff is<b>&nbsp;Σ<sub>1</sub></b>&nbsp;iff it is logically equivalent to a strictly&nbsp;Σ<sub>1<b>&nbsp;</b></sub>wff.</li><li>A wff is&nbsp;<b>strictly Π<sub>1</sub></b>&nbsp;iff it is of the form&nbsp;∀α∀β...∀ω φ, where&nbsp;φ is&nbsp;Δ<sub>0</sub>&nbsp;and&nbsp;α, β, ... , ω are one or more distinct variables that appear free in&nbsp;φ.</li><li>A wff is<b>&nbsp;Π<sub>1</sub></b>&nbsp;iff it is logically equivalent to a strictly Π<sub>1<b>&nbsp;</b></sub>wff.</li></ol><div>Note that these sets contain not only sentences but also open wff's, that is, wff's with one or more free variables. Here are two examples of Δ<sub>0</sub>&nbsp;wff's:</div><div><br /></div><div><div style="text-align: center;">(∃v ≤ x) (2 × v = x)&nbsp;</div></div><div><br /><div style="text-align: center;">2&nbsp;≤&nbsp;x &nbsp;∧&nbsp;&nbsp;(∀t ≤ x)&nbsp;(∀u ≤ x) (t&nbsp;× u = x&nbsp;→ (t = 1&nbsp;∨&nbsp;u = 1))</div><div style="text-align: center;"><br /></div>Each one of these two wff's contains the free variable x. Let's refer to these open wff's as e(x) and p(x), since they express the facts that x is even and x is prime, respectively.<br /><br /></div><div>According to Smith, the set&nbsp;Δ<sub>0</sub>&nbsp;is built in such a way that its elements do not contain any (unbounded) universal or existential quantifiers. I find this statement a bit surprising because the atomic&nbsp;Δ<sub>0</sub>&nbsp;wff of the form σ ≤ τ &nbsp;is&nbsp;really an abbreviation for:</div><div><br /></div><div style="text-align: center;">&nbsp;∃v (v +&nbsp;σ = τ)</div><div><br /></div><div>which <i>does</i> contain an unbounded existential quantifier. The only reasoning I could come up with to resolve this difficulty is that,&nbsp;to determine the truth value of&nbsp;σ ≤ τ,&nbsp;one only needs to check a finite number of addition facts, because the value of v that works, if it exists, must be less than or equal to the numerical value of&nbsp;τ (which must be a constant number for the truth value of the whole formula to be defined).<br /><br />Now, let's consider the following wff:<br /><br /><div style="text-align: center;">∀x <span class="Apple-style-span" style="font-size: large;">(</span> (e(x)&nbsp;∧ 4&nbsp;≤ x)&nbsp;→ (∃y ≤ x)&nbsp;(∃z ≤ x) (p(y)&nbsp;∧ p(z)&nbsp;∧ y + z = x)&nbsp;<span class="Apple-style-span" style="font-size: large;">)</span></div><br />The interpretation of this open wff is "Every even number greater than 2 is the sum of two primes," which is the famous Goldbach conjecture. Since the sub-formula after the&nbsp;∀x is a&nbsp;Δ<sub>0</sub>&nbsp;wff, the entire wff is strictly<b>&nbsp;</b>Π<sub>1</sub>. Recall that a&nbsp;strictly<b>&nbsp;</b>Π<sub>1&nbsp;</sub>wff is a&nbsp;Δ<sub>0</sub>&nbsp;wff preceded by a single group of one or more (unbounded) universal quantifiers. Similarly,&nbsp;a&nbsp;strictly<b>&nbsp;</b>Σ<sub>1&nbsp;</sub>wff is a&nbsp;Δ<sub>0</sub>&nbsp;wff preceded by a single group of one or more (unbounded) existential quantifiers.</div><br />Second, let's state some simple results: <br /><ul><li><b>Lemma 1:</b> The negation of a<b>&nbsp;</b>Δ<sub>0</sub>&nbsp;wff is also&nbsp;Δ<sub>0</sub>.</li><li><b>Lemma 2:</b> The negation of a<b>&nbsp;</b>Σ<sub>1&nbsp;</sub>wff&nbsp;is&nbsp;Π<sub>1</sub>&nbsp;and vice versa.</li><li><b>Lemma 3:</b>&nbsp;A&nbsp;Δ<sub>0</sub>&nbsp;wff is also both Σ<sub>1 </sub>and Π<sub>1</sub>.</li><li><b>Lemma 4:</b>&nbsp;The function that assigns a truth value to each&nbsp;Δ<sub>0</sub>&nbsp;wff is effectively computable.</li></ul>Lemma 1 follows directly from the second bullet point in the second definition above.<br /><br />Lemma 2 follows directly from De Morgan's laws for quantifiers, namely: <br /><ul><li>¬∃x&nbsp;φ(x) is logically equivalent to&nbsp;∀x&nbsp;¬φ(x)</li><li>¬∀x&nbsp;φ(x) is logically equivalent to&nbsp;∃x&nbsp;¬φ(x)</li></ul><br />Proof sketch of Lemma 3: <br /><ul><li>Let&nbsp;φ be any&nbsp;Δ<sub>0</sub>&nbsp;wff in which the variable z does not appear free.</li><li>φ&nbsp;is logically equivalent to both ∀z (φ&nbsp;∧&nbsp;z = z) and&nbsp;∃z (φ&nbsp;∧&nbsp;z = z).</li><li>∃z (φ&nbsp;∧&nbsp;z = z) is strictly&nbsp;Σ<sub>1</sub>.</li><li>∀z (φ&nbsp;∧&nbsp;z = z) is strictly&nbsp;Π<sub>1</sub>.</li></ul><br />Proof sketch of Lemma 4: <br /><ul><li>Computing the truth value of any&nbsp;Δ<sub>0</sub>&nbsp;wff takes only a finite number of well-defined steps because:</li><ul><li>Each universally or existentially bounded quantified wff can be converted to a finite conjunction or disjunction, respectively.</li><li>As mentioned above, any atomic&nbsp;Δ<sub>0</sub>&nbsp;wff using&nbsp;≤ can be converted to a finite disjunction.</li><li>One can use the truth tables of the connectives to compute truth values compositionally.</li></ul><li>This proof can be made rigorous using structural induction, where each atomic wff has a structural complexity of 0 and each connective increments the structural complexity by 1.&nbsp;</li></ul><br />Third, we come to the main result of this post: <br /><div><br /></div><div style="text-align: center;"><b>Theorem:</b>&nbsp;Q is Σ<sub>1</sub>-complete.&nbsp;</div><div style="text-align: center;"><br /></div>Proof sketch: <br /><ul><li>Q correctly decides every atomic&nbsp;Δ<sub>0</sub>&nbsp;sentence: these sentences are either closed equalities or closed inequalities; the first case is already covered in <a href="http://summerofgodel.blogspot.com/2013/06/baby-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">our proof that BA correctly decides all sentences in <i>L<sub>B</sub></i></span></a>; the second case also works out because each inequality between terms can be reduced to an inequality between numerals (since each closed term can be reduced to a numeral SSS...S0), which in turn is correctly decided, because Q captures the&nbsp;≤ relation.</li><li>Q correctly decides every Δ<sub>0</sub>&nbsp;sentence: we can again use structural induction here</li><li>Q proves every true Σ<sub>1&nbsp;</sub>sentence: since each true bounded kernel (that is, every true&nbsp;Δ<sub>0</sub>&nbsp;sentence) is proved by Q, every true sentence obtained by adding one or more existential quantifiers in front of the kernel is derivable in Q, using the inference rule called existential generalization (or existential introduction).</li></ul><div><br /></div><div>Finally, let's wrap up this post with a couple of corollaries.</div><div><br /></div><div style="text-align: center;"><b>Corollary 1:</b> &nbsp;A&nbsp;Π<sub>1</sub>&nbsp;sentence&nbsp;φ is true iff&nbsp;¬φ cannot be derived in Q (i.e.,&nbsp;φ is consistent with Q).</div><br />Proof sketch: <br /><ul><li>If&nbsp;φ is false, then&nbsp;¬φ is a true&nbsp;Σ<sub>1</sub>&nbsp;sentence (by Lemma 2) and thus derivable in Q.</li><li>If&nbsp;φ is true, then&nbsp;¬φ is false and thus not derivable in Q (since Q is sound, because its axioms are true and its inference rules are truth-preserving).</li></ul><div>So, if derivability in Q were effectively decidable (we'll see later that it's not), then the function that assigns a truth value to each sentence in&nbsp;Π<sub>1</sub>&nbsp;would be effectively computable.<br /><br />Before the next and last corollary, a quick definition:<br /><br />A theory T<sub>2</sub> <b>extends</b> a theory T<sub>1</sub> if: <br /><ul><li>The language of&nbsp;T<sub>1</sub>&nbsp;is a subset of the language of&nbsp;T<sub>2</sub>,</li><li>The wff's common to&nbsp;T<sub>1&nbsp;</sub>and T<sub>2</sub>&nbsp;have the same truth values in both interpreted languages, and</li><li>T<sub>2</sub>&nbsp;can prove (at least) all of the theorems of&nbsp;T<sub>1</sub>.</li></ul><br /><div style="text-align: center;"><b>Corollary 2:</b>&nbsp;&nbsp;If a theory T extends Q,&nbsp;then T is consistent iff T is&nbsp;Π<sub>1</sub>-sound.</div><br />Proof sketch: Assume T extends Q.<br /><ul><li>If T is inconsistent, then it can derive everything in T's language, including all of the false&nbsp;Π<sub>1</sub>&nbsp;sentences. Therefore, T is not&nbsp;Π<sub>1</sub>-sound.</li><li>If T is&nbsp;not Π<sub>1</sub>-sound, then T proves some false&nbsp;Π<sub>1</sub>&nbsp;sentence&nbsp;φ. Thus,&nbsp;¬φ is a true&nbsp;Σ<sub>1</sub>&nbsp;sentence (by Lemma 2) and T must derive&nbsp;¬φ (since Q being&nbsp;Σ<sub>1</sub>-complete implies that T is also&nbsp;Σ<sub>1</sub>-complete). Therefore T is inconsistent (since it derives both&nbsp;φ and&nbsp;¬φ). By contraposition, if T is consistent, it is&nbsp;Π<sub>1</sub>-sound.</li></ul></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/h6bjDny76z8" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-is-1-complete.htmltag:blogger.com,1999:blog-6792710671733445593.post-59951858489227457262013-07-07T17:03:00.004-07:002013-07-07T17:04:47.870-07:00Robinson Arithmetic is order-adequateIn chapter 9, <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372464710&amp;sr=1-2" target="_blank"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> defines the following concept: A theory T that <a href="http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-captures-less-than.html" target="_blank"><span class="Apple-style-span" style="color: blue;">captures the relation&nbsp;≤</span></a>&nbsp;is <b>order-adequate</b> if it satisfies the following nine properties:<br /><ul><li>O1:&nbsp;T ⊢ ∀x (0 ≤ x)</li><li>O2:&nbsp;For any n, T ⊢ ∀x ((x = 0 ∨&nbsp;x = 1 ∨&nbsp;x = 2 ∨ ...&nbsp;∨&nbsp;x =&nbsp;n) → x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)</li><li>O3:&nbsp;For any n, T ⊢ ∀x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;(x = 0 ∨&nbsp;x = 1 ∨&nbsp;x = 2 ∨ ...&nbsp;∨&nbsp;x =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>))</li><li>O4: For any n, if T ⊢ φ(0) and&nbsp;T ⊢ φ(1) and ... and&nbsp;T ⊢ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;then&nbsp;T ⊢ (∀x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)φ(x)</li><li>O5: For any n, if T ⊢ φ(0) or T ⊢ φ(1) or ... or T ⊢ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;then&nbsp;T ⊢ (∃x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)φ(x)</li><li>O6:&nbsp;For any n, T ⊢ ∀x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;x ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)</li><li>O7:&nbsp;For any n, T ⊢ ∀x (<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;≤ x&nbsp;&nbsp;→ (<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;= x &nbsp;∨&nbsp;S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;≤ x))</li><li>O8:&nbsp;For any n, T ⊢ ∀x&nbsp;(x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;∨&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;≤ x)</li><li>O9:&nbsp;For any n&gt;0, T ⊢ (∀x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n-1</span>)φ(x)&nbsp;→&nbsp;(∀x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)(x ≠&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;φ(x))</li></ul><div>Then, we have the following theorem: <br /><a name='more'></a><br /><div style="text-align: center;"><a href="http://summerofgodel.blogspot.com/2013/06/q-robinson-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Robinson Arithmetic (or Q)</span></a> is order-adequate.</div><div style="text-align: center;"><br /></div>Smith proves that Q satisfies a few of the 9 properties above, namely O1, O2, O3, and O8. Below, I attempt to prove a few of the other properties.<br /><br /></div><div>Proof of O4:&nbsp;Assume that n is an arbitrary natural number.</div><div><ol><li>Assume that a is an arbitrary term of Q such that a ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span></li><li>Using O3:&nbsp;∀x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;(x = 0 ∨&nbsp;x = 1 ∨&nbsp;x = 2 ∨ ...&nbsp;∨&nbsp;x =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>))</li><li>a ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;(a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>) [by universal instantiation]</li><li>a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span> [modus ponens 1&amp;3]</li><li>Sub-proof by cases:&nbsp;</li><ul><li>According to 4, a is equal to one of the terms in {0,1,2,...,&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>}</li><li>In all cases, Q ⊢ φ(a) [hypothesis of O4]</li><li>Therefore Q ⊢ φ(a)</li></ul><li>Q ⊢&nbsp;a ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→ φ(a) [discharging assumption 1]</li><li>Q ⊢&nbsp;∀x&nbsp;(x&nbsp;≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→ φ(x)) [by universal generalization]</li><li>Q ⊢ (∀x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;φ(x) [by definition of the bounded universal quantifier]</li></ol><div><br /><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Proof of O5:&nbsp;Assume that n is an arbitrary natural number.</div></div><div><ol><li>There is at least one numeral a such that&nbsp;Q ⊢&nbsp;φ(a) [hypothesis of O5]</li><li>Q ⊢&nbsp;a&nbsp;≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[since 'the number denoted by a'&nbsp;≤ n&nbsp;and Q captures&nbsp;≤]</li><li>Q ⊢&nbsp;a&nbsp;≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;∧&nbsp;φ(a) [conjunction of 1 and 2]</li><li>Q ⊢ ∃x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;∧&nbsp;φ(x)) [existential generalization]</li><li>Q ⊢ (∃x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;φ(x)&nbsp;[by definition of the bounded existential quantifier]</li></ol></div></div></div><div><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Proof of O6:&nbsp;Assume that n is an arbitrary natural number.</div></div><div><ol><li>Assume (inside Q) that a is an arbitrary term such that a ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span></li><li>Using O3:&nbsp;∀x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;(x = 0 ∨&nbsp;x = 1 ∨&nbsp;x = 2 ∨ ...&nbsp;∨&nbsp;x =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>))</li><li>a ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;(a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>) [by universal instantiation]</li><li>a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[modus ponens 1&amp;3]</li><li>a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;∨ a = S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[added disjunct to 4]</li><li>Using O2:&nbsp;&nbsp;∀x ((x = 0 ∨&nbsp;x = 1 ∨&nbsp;x = 2 ∨ ...&nbsp;∨&nbsp;x =&nbsp;n&nbsp;∨&nbsp;x = Sn)&nbsp;→ x ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)</li><li>(a = 0 ∨ a = 1 ∨ a = 2 ∨ ...&nbsp;∨ a =&nbsp;n&nbsp;∨ a = S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;→ a ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[by universal instantiation]</li><li>a ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[modus ponens 5&amp;7]</li><li>a&nbsp;≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;a ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;[discharging assumption 1]</li><li>∀x (x ≤&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;→&nbsp;x ≤ S<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>) [universal generalization]</li></ol><div><div><ol><ol></ol></ol></div></div></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/FJzRfV-wSV4" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-is-order-adequate.htmltag:blogger.com,1999:blog-6792710671733445593.post-57847669726595089822013-07-03T19:27:00.001-07:002013-07-03T19:27:06.293-07:00Robinson Arithmetic captures the "less-than-or-equal-to" relationIn this post, we'll start discussing the material in Chapter 9 of <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372464710&amp;sr=1-2" target="_blank"><span class="Apple-style-span" style="color: blue;">Peter Smith's book</span></a>, namely up to section 9.3.<br /><br />Before proceeding, review the definition of <a href="http://summerofgodel.blogspot.com/2013/06/q-robinson-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Robinson Arithmetic</span></a>&nbsp;(denoted Q) as well as what it means for a theory to <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;">capture a numerical relation</span></a>.<br /><br />Now, we'll show that Robinson Arithmetic captures the ≤ numerical relation with the open well-formed formula:&nbsp;∃x&nbsp;(x + m = n).<br /><br />In other words, we are going to prove the following two-part theorem:<br /><blockquote class="tr_bq"><b>Theorem:</b> If m and n are natural numbers, then:<br />1. If m&nbsp;≤ n, then Q ⊢&nbsp;∃x (x + <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span> = <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)<br />2. If m &gt; n, then Q ⊢&nbsp;¬&nbsp;∃x (x + <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span> = <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>) </blockquote><div>Recall that if m and n are natural numbers, then&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>&nbsp;and&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;are the numerical terms representing m and n, respectively, in the formal theory.<br /><br /><u>Proof sketch of part 1:</u><br /><a name='more'></a></div><div><ul><li>Assume that&nbsp;m&nbsp;≤ n</li><li>There must exist a natural number k such that: k + m = n</li><li>Since Q can prove everything that <a href="http://summerofgodel.blogspot.com/2013/06/baby-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">BA</span></a> proves, and BA can prove all true equations, we have: Q&nbsp;⊢ <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">k</span> +&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>&nbsp;=&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span></li><li>By existential generalization (an inference rule also called 'existential quantifier introduction'), we have: Q&nbsp;⊢&nbsp;∃x (x&nbsp;+&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>&nbsp;=&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)</li></ul><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Now, here are axioms of Q that we will need in the second part of the proof: <br /><ul><li>Axiom 1: ∀x (0 ≠ Sx)</li><li>Axiom 2: ∀x∀y&nbsp;(Sx = Sy → x = y)</li><li>Axiom 4: ∀x (x + 0 = x)</li><li>Axiom 5: ∀x∀y&nbsp;(x + Sy = S(x + y))</li></ul></div><u><br /></u><u>Proof sketch of part 2:</u><br /><div><ul><li>Assume that m &gt; n. We need to prove&nbsp;Q ⊢&nbsp;¬&nbsp;∃x (x +&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>&nbsp;=&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)</li><li>Following Smith's lead, we will only write down a formal proof for the special case m = 2 and n = 1 here, hoping that the general proof pattern will be easily discernible. Note that Smith uses a Fitch-style, natural-deduction proof, while I came up with a Hilbert-style proof. Here is the formal proof of&nbsp;¬ ∃x&nbsp;(x + SS0 = S0) in Q:</li></ul><div></div><ol><li>∀x (0 ≠ Sx) [Axiom 1]</li><li>0 ≠ Sa [Universal instantiation of 1 with arbitrary term a]</li><li>∀x∀y&nbsp;(Sx = Sy → x = y) [Axiom 2]</li><li>∀y&nbsp;(S0 = Sy → 0 = y)&nbsp;[Universal instantiation of 3 with specific term 0]</li><li>S0 = SSa → 0 = Sa&nbsp;[Universal instantiation of 4 with term Sa]</li><li>0 ≠ Sa&nbsp;→&nbsp;S0 ≠ SSa [Contrapositive of 5]</li><li>S0 ≠ SSa [Modus ponens 2 &amp; 6]</li><li>∀x (x + 0 = x) [Axiom 4]</li><li>a + 0 = a [Universal instantiation of 8 with term a]</li><li>S0 ≠ SS(a+0) [Substitution of (a+0) for a in 7]</li><li>∀x∀y&nbsp;(x + Sy = S(x + y)) [Axiom 5]</li><li>∀y&nbsp;(a + Sy = S(a + y))&nbsp;[Universal instantiation of 11 with term a]</li><li>a + S0 = S(a + 0)&nbsp;[Universal instantiation of 12 with term 0]</li><li>S0 ≠ S(a + S0) [Substitution of (a + S0) for S(a + 0) in 10]</li><li>a + SS0 = S(a + S0)&nbsp;[Universal instantiation of 12 with term S0]</li><li>S0 ≠ a + SS0&nbsp;[Substitution of a + SS0 for S(a + S0) in 14]</li><li>∀x (S0 ≠ a + SS0) [Universal generalization of 16]</li><li>∀x ¬ (S0 = x + SS0) [Rewriting of 17]</li><li>¬ ∃x&nbsp;(S0 = x + SS0) [Apply one of De Morgan's law for quantifiers to 18]</li><li>¬ ∃x&nbsp;(x + SS0 = S0) [Switched order of terms in equation 19]</li></ol></div><div>Since Q captures the ≤ relation, we can now add this predicate to <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;"><i>L<sub>A</sub></i> (the interpreted, formal language of Q)</span></a>. In other words, we can use the abbreviation α ≤ β for&nbsp;∃x (x + α = β). <br /><br />In fact, we will use a couple more abbreviations for <b>bounded quantification</b>, namely: <br /><ul><li>(∀x ≤ k) φ(x) as syntactic sugar for&nbsp;∀x (x ≤ k → φ(x)), and</li><li>(∃x ≤ k) φ(x) as syntactic sugar for&nbsp;∃x (x ≤ k ∧ φ(x))</li></ul><br />In conclusion, to unpack the two levels of abbreviation: <br /><ul><li>(∀x ≤ k) φ(x) is syntactic sugar for&nbsp;∀x (∃y (y + x = k)&nbsp;→ φ(x))</li><li>(∃x ≤ k) φ(x) is syntactic sugar for&nbsp;∃x (∃y (y + x = k)&nbsp;∧ φ(x))</li></ul></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/lhfuVRblTk4" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/07/robinson-arithmetic-captures-less-than.htmltag:blogger.com,1999:blog-6792710671733445593.post-75199324335618058252013-06-30T16:00:00.001-07:002013-06-30T16:01:27.688-07:00Q - Robinson ArithmeticNow that we are familiar with <a href="http://summerofgodel.blogspot.com/2013/06/baby-arithmetic.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Baby Arithmetic (BA)</span></a>, we can make its language more expressive by allowing variables and quantifiers back into its logical vocabulary. When we do this, we simply obtain the interpreted language&nbsp;<i>L<sub>A</sub></i>, that was <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;">described earlier</span></a>.<br /><br />Since we now have variables and quantifiers, we can replace the schemata of BA with regular axioms (see below). The resulting formal system of arithmetic is called <b>Robinson Arithmetic</b>&nbsp;and is often denoted by the letter <b>Q</b>, as described in Chapter 8 of <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372464710&amp;sr=1-2" target="_blank"><span class="Apple-style-span" style="color: blue;">Peter Smith's book</span></a>. Here is his definition of Q:<br /><a name='more'></a><ul><li>The interpreted language of Robinson Arithmetic is simply the<a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;"> language&nbsp;<i>L<sub>A</sub></i>,&nbsp;together with the interpretation&nbsp;<i>I<sub>A</sub></i></span></a>.&nbsp;</li><li>The deductive apparatus (inference rules) of Robinson Arithmetic will be some version of first-order logic with identity.&nbsp;We'll settle on a specific logic in a future post.</li><li>The axioms of Robinson Arithmetic are:</li><ul><li>Axiom 1: ∀x (0 ≠ Sx)</li><li>Axiom 2: ∀x∀y&nbsp;(Sx = Sy → x = y)</li><li>Axiom 3: ∀x (x&nbsp;≠ 0&nbsp;→ ∃y (x = Sy))</li><li>Axiom 4: ∀x (x + 0 = x)</li><li>Axiom 5: ∀x∀y&nbsp;(x + Sy = S(x + y))</li><li>Axiom 6: ∀x (x × 0 = 0)</li><li>Axiom 7: ∀x∀y&nbsp;(x&nbsp;×&nbsp;Sy = (x&nbsp;×&nbsp;y) + x)</li></ul></ul><div>Each axiom above, except for axiom 3, is a direct rewriting in first-order logic of a schema in the<a href="http://summerofgodel.blogspot.com/2013/06/baby-arithmetic.html" target="_blank"> <span class="Apple-style-span" style="color: blue;">definition of BA</span></a>. Axiom 3 is added to make sure that there is no element besides zero that does not have a successor.</div><div><br /></div><div>Then Smith proves the following theorem:</div><div><br /></div><div style="text-align: center;">&nbsp;Q is not negation-complete.</div><div style="text-align: center;"><br /></div><div>Proof sketch: Let φ be the sentence ∀x (0 + x = x)</div><div><ul><li>Q is sound, since its axioms are all true and its logic is truth-preserving.</li><li>Q cannot derive φ:&nbsp;</li><ul><li>One way to prove this fact is to come up with an alternative interpretation for&nbsp;<i>L<sub>A</sub></i>&nbsp;that makes Q's axioms true but&nbsp;φ false. Since Q is sound, it cannot derive&nbsp;φ. (Note: The alternative interpretation used in Smith's proof is an extension of&nbsp;<i>I<sub>A</sub></i>&nbsp;with two new elements&nbsp;<i>a</i>&nbsp;and&nbsp;<i>b</i>&nbsp;in the domain such that&nbsp;<i>a</i>&nbsp;is its own successor,&nbsp;<i>b</i>&nbsp;is its own successor, and addition and multiplication are extended in such a way that 0 +&nbsp;<i>a</i>&nbsp;is equal to&nbsp;<i>b&nbsp;</i>and 0 +&nbsp;<i>b</i>&nbsp;is equal to&nbsp;<i>a</i>, thereby falsifying&nbsp;φ.)<div></div></li></ul><li>Q cannot derive ¬ φ:</li><ul><li>This is because&nbsp;Q is sound and&nbsp;¬ φ is false under the original interpretation&nbsp;<i>I<sub>A</sub></i>.</li></ul></ul></div><div>So this chapter contains yet another proof of incompleteness that is much simpler than Gödel's proof.<br /><br />Since&nbsp;φ represents a trivial fact about addition that Q cannot prove, it must be the case that Q is a weak theory of arithmetic.<br /><br />Nevertheless, Q is interesting because Smith promises to prove (in a future chapter) that Q is <a href="http://summerofgodel.blogspot.com/2013/06/consistent-sufficiently-strong-formal.html" target="_blank"><span class="Apple-style-span" style="color: blue;">sufficiently strong</span></a>. Recall that this property is defined in terms of <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html" target="_blank"><span class="Apple-style-span" style="color: blue;">capturing properties and relations using <i>case-by-case</i> proofs.</span></a><br /><br />And, sure enough, Q can do that. For example, Q <i>can</i> derive all of the sentences 0 + 0 = 0,&nbsp;0 + S0 = S0,&nbsp;0 + SS0 = SS0, etc. It simply cannot derive the universally quantified sentence&nbsp;φ that covers this infinite set in one fell swoop.<br /><br />Finally (for this chapter anyway), Smith points out that any consistent theory that extends Q will also be sufficiently strong and therefore negation incomplete.<br /><br />The next chapter presents Robinson Arithmetic in much greater detail and will probably take us several posts to dissect.</div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/jP9HQ5vnhZA" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/q-robinson-arithmetic.htmltag:blogger.com,1999:blog-6792710671733445593.post-71198609351377435022013-06-28T20:07:00.002-07:002013-06-28T20:07:45.349-07:00Baby arithmeticSince chapter 7 in <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372464710&amp;sr=1-2"><span class="Apple-style-span" style="color: blue;">Peter Smith's book</span></a> is so short, I'll summarize it quickly and then start discussing chapter 8.<br /><br />Chapter 7 starts by comparing the two incompleteness theorems discussed in previous chapters (let's call these "Smith's theorems") to Gödel's first incompleteness theorem as follows:<br /><a name='more'></a><ol><li>Each one of Smith's theorems states that formal systems that satisfy two conditions are incomplete. In <a href="http://summerofgodel.blogspot.com/2013/06/sound-formal-systems-with-sufficiently.html"><span class="Apple-style-span" style="color: blue;">Smith's first theorem</span></a>, the formal system must be sound and its language must be sufficiently expressive.&nbsp;In <a href="http://summerofgodel.blogspot.com/2013/06/consistent-sufficiently-strong-formal.html"><span class="Apple-style-span" style="color: blue;">Smith's second theorem</span></a>, the formal system must be consistent (a syntactic property that is weaker than soundness, a semantic property) and sufficiently strong (a property that is stronger than sufficient expressiveness). Smith states that Gödel's first theorem also comes in two varieties, with a similar trade-off between the strengths of two conditions.</li><li>Unlike Gödel's proof, the proofs of Smith's theorems do not involve <a href="http://summerofgodel.blogspot.com/2013/06/proof-sketch-of-godels-incompleteness.html"><span class="Apple-style-span" style="color: blue;">self-reference</span></a> through the <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-arithmetization-of-meta.html"><span class="Apple-style-span" style="color: blue;">arithmetization of meta-mathematics</span></a>. On the other hand, Gödel's proof is constructive: it includes a method for building a true sentence that is unprovable. Smith's proofs are much easier; but they are not constructive.</li><li>Gödel's proof does not rely on the whole class of <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">decidable numerical properties and relations</span></a> as Smith's proofs do, through the definitions of <a href="http://summerofgodel.blogspot.com/2013/06/sound-formal-systems-with-sufficiently.html"><span class="Apple-style-span" style="color: blue;">sufficient expressiveness</span></a> and <a href="http://summerofgodel.blogspot.com/2013/06/consistent-sufficiently-strong-formal.html"><span class="Apple-style-span" style="color: blue;">sufficient strength</span></a>. Instead,&nbsp;Gödel's proof&nbsp;relies on the class of so-called&nbsp;<i>primitive recursive</i> properties and relations, which is a strict subset of the decidable ones.</li></ol><div>The rest of Chapter 7 is a road map for the next 10 chapters or so, which culminate in the proof of Gödel's first theorem, but start with two well-known formal systems of arithmetic,&nbsp;namely "Robinson Arithmetic" and "First-Order Peano Arithmetic,"&nbsp;which are both based on the <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">formal language&nbsp;<i>L<sub>A</sub></i></span>,</a><br /><br />But first, Smith describes a simpler formal system, called <b>Baby Arithmetic</b> (or <b>BA</b>), which is the part of Chapter 8 that I want to discuss in the rest of this post.<br /><ul><li>The language of BA, let's call it <i><b>L<sub>B</sub></b></i>, is a strict subset of the language <i>L<sub>A</sub></i>. They have the same non-logical vocabulary, namely the symbols 0, S, + and&nbsp;×. However, the logical vocabulary of&nbsp;<i>L<sub>B</sub></i>&nbsp;is missing numerical variables and quantifiers (it still contains the equality symbol and the usual connectives). The interpretation of&nbsp;<i>L<sub>B</sub></i>&nbsp;is identical to that of the corresponding components of&nbsp;<i>L<sub>A</sub></i>.</li><li>The deductive apparatus of BA can include any of the usual inference rules of propositional logic, together with rules for the equality predicate; for example, from any sentence φ and the identity ρ = τ, where ρ and τ are terms, <i>Leibniz's law</i>&nbsp;enables us to infer the sentence obtained by substituting in φ every occurrence of ρ by τ.</li><li>BA contains 6 non-logical axioms that specify the structure of the natural number sequence and define the addition and multiplication functions. Since BA does not contain quantifiers nor variables, each axiom is a template or <b>schema</b> using the Greek letters α and β as placeholders for numerical expressions. So each schema generates an infinite number of sentences, one for each possible substitution of the Greek letter(s). For example, the first schema below generates the sentences&nbsp;0 ≠ S0,&nbsp;0 ≠ SS0,&nbsp;0 ≠ SSS0,&nbsp;0 ≠ S(S0&nbsp;× SS0),&nbsp;etc.&nbsp;Here are the 6 axiom schemata of BA:</li><ul><li>Schema 1:&nbsp;0 ≠ Sα</li><ul><li>i.e., 0 is not the successor of any number</li></ul><li>Schema 2: (Sα = Sβ) → (α = β)</li><ul><li>i.e., by contraposition, any two distinct numbers have distinct successors</li></ul><li>Schema 3: α + 0 = α</li><ul><li>i.e., base case for the definition of addition</li></ul><li>Schema 4: α + Sβ = S(α + β)</li><ul><li>i.e., recursive case for the definition of addition</li></ul><li>Schema 5: α&nbsp;×&nbsp;0 = 0</li><ul><li>i.e., base case for the definition of multiplication</li></ul><li>Schema 6: α&nbsp;×&nbsp;Sβ = (α&nbsp;×&nbsp;β) +&nbsp;α</li><ul><li>i.e., recursive case for the definition of multiplication</li></ul></ul></ul><div>Note that BA meets all of the requirements of an <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">axiomatized formal theory</span></a> (or AFT). For example, it is decidable whether or not a sentence of&nbsp;&nbsp;<i>L<sub>B</sub></i>&nbsp;is an instance of an axiom schema of BA.</div><div><br /></div><div>Then, Smith proves that BA is negation-complete, that is, for any sentence φ in&nbsp;<i>L<sub>B</sub></i>, either BA&nbsp;⊢ φ or BA ⊢ ¬ φ. In fact, Smith proves that BA <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">correctly decides</span></a> all sentences in&nbsp;<i>L<sub>B</sub></i>.<br /><br />Finally, since BA is negation-complete, we can automatically infer, using <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">a previously discussed theorem</span></a>, that BA is decidable.</div><br />In conclusion, BA is complete: it can prove every true statement that its language can express. Unfortunately, its language can express little. It can express specific statements, such as: 2 + 3 = 5, 4 ≠ 2 × 3, the successor of 0 is not 0, the successor of 1 is not 0, the successor of 2 is not 0, etc. However, it cannot express general statements, such as: all natural numbers have a successor that is different from 0. Nevertheless, stronger AFTs of arithmetic can be obtained by extending BA.</div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/vEopGBUGjos" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/baby-arithmetic.htmltag:blogger.com,1999:blog-6792710671733445593.post-32570795668280325702013-06-27T19:39:00.000-07:002013-06-27T19:39:39.741-07:00Consistent, sufficiently strong formal systems of arithmetic are incompleteIn Chapter 6, <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372380122&amp;sr=1-2"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> proves another incompleteness theorem. To place this theorem in context, let's review some definitions about <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">axiomatized formal theories</span></a> (or AFTs) from earlier posts.<br /><br />Let T be some AFT.<br /><ul><li>T is <b>consistent</b> iff (if and only if) there is no sentence&nbsp;φ such that T proves both φ and ¬ φ.</li><li>T is <b>sound</b> iff every theorem that T proves is true according to the interpretation that is built into T's language.</li><li>T is <b>(negation-)complete</b> iff for every sentence&nbsp;φ in T's language, T&nbsp;proves either φ or ¬ φ.</li><li>T is <b>decidable</b> iff there is an algorithm that, given any sentence&nbsp;φ in T's language, determines whether or not T proves&nbsp;φ.</li><li>If needed, go <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">here</span></a> to review what it means for T's language to <b>express</b> a numerical property P or a numerical relation R.</li><li>If needed, go&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">here</span></a>&nbsp;to review what it means for T to&nbsp;<b>capture</b>&nbsp;a numerical property P or a numerical relation R.&nbsp;</li><li>If needed, go&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/sound-formal-systems-with-sufficiently.html"><span class="Apple-style-span" style="color: blue;">here</span></a>&nbsp;to review what it means for T's language to <b>be sufficiently </b><b>expressive</b>.</li></ul><div><br />We'll use the following abbreviations: <br /><a name='more'></a></div><br /><center><table border="1"><tbody><tr><th>The letter ...</th><th>stands for...</th></tr><tr><td><div style="text-align: center;">C</div></td><td>T is Consistent</td></tr><tr><td><div style="text-align: center;">S</div></td><td>T is Sound</td></tr><tr><td><div style="text-align: center;">N</div></td><td><div style="text-align: left;">T is Negation-complete</div></td></tr><tr><td><div style="text-align: center;">D</div></td><td><div style="text-align: left;">T is Decidable</div></td></tr><tr><td><div style="text-align: center;">E</div></td><td>T's language is sufficiently Expressive</td></tr><tr><td><div style="text-align: center;">G</div></td><td>T is sufficiently stronG</td></tr></tbody></table></center><br />The new material in this post deals with the new property that appears in the last row of this table. <br /><br />Recall that, in a <a href="http://summerofgodel.blogspot.com/2013/06/sound-formal-systems-with-sufficiently.html"><span class="Apple-style-span" style="color: blue;">previous post</span></a>, we showed that any AFT that is sound and sufficiently expressive must be negation-incomplete, or, written in pseudo-propositional logic (this should really be a universally quantified formula):<br /><blockquote class="tr_bq" style="text-align: center;"><b>Theorem 1</b>: (S ∧ E) → ¬ N</blockquote>Soundness, by itself, is not enough to ensure incompleteness. Neither is&nbsp;sufficient expressiveness. To ensure incompleteness, both properties are needed. However, it is possible to weaken one of these properties; and as long as we simultaneously strengthen the other property, we can still guarantee incompleteness.<br /><br />Notice that soundness is stronger than consistency. &nbsp;S → C holds because, if an AFT only proves true sentences, then it cannot prove both a sentence and its negation. However, consistency does not imply soundness. For example, an AFT could prove all of the false sentences and not be inconsistent, as long as it does not prove a single true sentence.<br /><br />So, we can weaken the S condition down to the C condition. However, to ensure incompleteness, we will strengthen the E condition to the G condition, yielding:<br /><br /><div style="text-align: center;"><b>Theorem 2</b>: (C ∧ G) → ¬ N</div><div style="text-align: center;"><br /></div><div style="text-align: left;">All that is left to do is define the property G and give a proof sketch for this theorem.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">We say that an AFT of arithmetic is <b>sufficiently strong</b> iff it captures all <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively decidable numerical properties</span></a>. Recall that <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">capturing a property is stronger than expressing a property</span></a>.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Now, to prove theorem 2, we can use the following two lemmata:</div><div style="text-align: left;"><br /></div><div style="text-align: center;"><b>Lemma 1:</b>&nbsp;(C ∧ N) → D</div><div style="text-align: center;"><br /></div><div style="text-align: left;">which was proved in <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">an earlier post</span></a>, and</div><div style="text-align: center;"><br /></div><div style="text-align: center;"><b>Lemma 2:</b>&nbsp;(C ∧ G) → ¬ D</div><div style="text-align: center;"><br /></div><div style="text-align: left;">which states that "no consistent, sufficiently strong AFT is decidable." One way to prove this lemma is to assume that C, G and D are all true and exhibit a contradiction using Cantor's diagonalization method (a proof of this type was discussed in&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/an-enumerable-but-not-effectively.html"><span class="Apple-style-span" style="color: blue;">this post</span></a>).</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Finally, we can easily write a formal proof in propositional logic of Theorem 2, that is, N is false under the assumption that C ∧ G is true.</div><div style="text-align: left;"></div><ol><li>C ∧ G [Assumption]</li><li>(C ∧ G) → ¬ D [Lemma 2]</li><li>¬ D [Modus ponens applied to 1 and 2]</li><li>&nbsp;(C ∧ N) → D [Lemma 1]</li><li>¬ D&nbsp;→&nbsp;¬&nbsp;(C ∧ N) [Contraposition of 4]</li><li>¬&nbsp;(C ∧ N)&nbsp;[Modus ponens applied to 3 and 5]</li><li>¬ C &nbsp;∨ ¬&nbsp;N [one of De Morgan's laws]</li><li>C → ¬ N [logical equivalence applied to 7]</li><li>C [simplification of 1]</li><li>¬&nbsp;N [Modus ponens applied to 8 and 9]</li></ol><div>So, from&nbsp;C ∧ G, we derived&nbsp;¬&nbsp;N, that is:</div><blockquote class="tr_bq" style="text-align: left;"><b>Theorem 2:</b> Consistent, sufficiently strong formal systems of arithmetic are incomplete.</blockquote><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/G5BRBSgnbTE" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/consistent-sufficiently-strong-formal.htmltag:blogger.com,1999:blog-6792710671733445593.post-55238091156552229392013-06-24T18:08:00.000-07:002013-06-27T19:43:02.210-07:00Sound formal systems with sufficiently expressive languages are incompleteIn chapter 5, <a href="http://www.amazon.com/Introduction-G%25f6dels-Cambridge-Introductions-Philosophy/dp/0521674530/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1372116675&amp;sr=1-2"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> uses a counting argument to prove that sound axiomatized formal theories (AFTs) that can express a good amount of arithmetic are incomplete. In short: since their set of theorems is effectively enumerable but their set of true sentences is not, the two sets must be different. In a sound theory, the theorems are all true. Therefore, some truths must remained unproved in such theories.<br /><br />Let's start with a definition. The language of an AFT of arithmetic is <b>sufficiently expressive</b> if and only if:<br /><ol><li>It can express quantification over numbers, and</li><li>It can <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">express</span></a> every <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively decidable two-place numerical relation</span></a>.</li></ol>Now, to the pivotal theorem of this chapter: <br /><a name='more'></a><blockquote class="tr_bq"><b>Theorem 1:</b>&nbsp;The set of true sentences of a sufficiently expressive language of arithmetic is not effectively enumerable.</blockquote>Proof sketch: <br /><ul><li><a href="http://summerofgodel.blogspot.com/2013/06/an-enumerable-but-not-effectively.html"><span class="Apple-style-span" style="color: blue;">According to the previous post</span></a>, the set K is effectively enumerable. Therefore, there is an effectively computable function f from&nbsp;<span class="Apple-style-span" style="font-family: 'Arial Unicode MS', 'Lucida Grande', sans-serif; font-size: 14px;">ℕ</span>&nbsp;to K that enumerates K.&nbsp;</li><li>Now, consider the binary relation <i>R(x,y)</i> that is true if and only if f(x)=y. Then, n ∈ K if and only if there exists a number x such that <i>R(x,n)</i>&nbsp;holds.</li><li><i><span class="Apple-style-span" style="font-style: normal;">Since f is effectively computable,&nbsp;</span>R</i> is effectively decidable; so, by definition, a sufficiently expressive (interpreted) language L can <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">express</span></a> <i>R</i> and the existential quantifier. Let&nbsp;∃x R(x,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)<i>&nbsp;</i>be a sentence of L that <a href="http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.html"><span class="Apple-style-span" style="color: blue;">expresses</span></a><i>&nbsp;"<span class="Apple-style-span" style="font-style: normal;">there exists a number x such that</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>R(x,n)</i></span><span class="Apple-style-span" style="font-style: normal;">&nbsp;holds.</span>"</i></li><li>In short: n&nbsp;∈ K if and only if&nbsp;∃x R(x,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true in L's interpretation.</li><li>That is: n&nbsp;∉ K if and only if&nbsp;¬ ∃x R(x,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true in L's interpretation.</li><li>If the set of true sentences of L were&nbsp;effectively enumerable, then we could go through and, for each formula of the form&nbsp;¬ ∃x R(x,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>), we could output the value of&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;(while skipping all of the other sentences). But this algorithm would constitute an effective enumeration of&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>. Since <a href="http://summerofgodel.blogspot.com/2013/06/an-enumerable-but-not-effectively.html"><span class="Apple-style-span" style="color: blue;">we proved that <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>&nbsp;is&nbsp;not effectively enumerable</span></a>, the set of true sentences of L is not effectively enumerable.</li></ul><br />The main result of this post follows easily from Theorem 1.<br /><br />Let's call S1 the set of theorems of an AFT built on top of a sufficiently expressive language L. <a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">We know</span></a> that S1 is effectively enumerable. But, according to theorem 1, the set S2 of the true sentences of L is not effectively enumerable. From this simple counting argument, we can infer that S1 ≠ S2. So, either some theorems are not true or some true sentences are not theorems.<br /><br />If we assume that the AFT is sound, then it must be the case that S1 ⊂ S2. Therefore, there must exist at least one true sentence φ in L that is not a theorem. In addition, since&nbsp;φ&nbsp;is true,&nbsp;¬ φ is false and thus cannot be a theorem either. In conclusion, neither φ nor ¬ φ is a theorem. This is a proof of:<br /><blockquote class="tr_bq"><b>Theorem 2:</b>&nbsp;Sound AFTs of arithmetic whose language is sufficiently expressive are negation-incomplete.</blockquote><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/6C3868WIAEA" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com1http://summerofgodel.blogspot.com/2013/06/sound-formal-systems-with-sufficiently.htmltag:blogger.com,1999:blog-6792710671733445593.post-73836368097988803872013-06-23T20:45:00.002-07:002013-06-23T20:45:32.794-07:00An enumerable but not effectively enumerable setLet's focus on the first half of chapter 5 in <a href="http://www.amazon.com/Introduction-G-f6dels-Cambridge-Introductions-Philosophy/dp/0521674530"><span class="Apple-style-span" style="color: blue;">Peter Smith's book</span></a>. First, Smith describes a new characterization of <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively enumerable sets</span></a>. Second, Smith gives an example of a set of integers that is not effectively enumerable.<br /><br />Let's define the <b>numerical domain</b> of an algorithm as the set of natural numbers with the following property: when the algorithm takes one of these numbers as input, it terminates and returns a result. Every algorithm has a numerical domain. If the input to some algorithm is not a single natural number, then the numerical domain of this algorithm is the empty set. Otherwise, the numerical domain is the set of those natural numbers on which the algorithm does not crash and does not go into an infinite loop.<br /><br />It turns out that each numerical domain is effectively enumerable. In fact, the converse is also true, according to the following theorem:<br /><a name='more'></a><blockquote class="tr_bq"><b>Theorem 1: </b>W is an effectively enumerable set of natural numbers if and only if W is the numerical domain of some algorithm.</blockquote>Proof sketch of the "only if" direction:<br /><ul><li>Let f be the <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively computable function</span></a> that enumerates W.</li><li>Build the algorithm that, when taking n as input, computes the values f(0), f(1), f(2), ..., f(i) and only stops if and when f(i) = n and then returns, say, 0.</li><li>This algorithm will only stop when its input is in W. Therefore, its numerical domain is W.</li></ul><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Proof sketch of the "if" direction:</div><ul><li>Let A be an algorithm with numerical domain W.</li><li>We know that&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">the set of pairs of natural numbers is effectively enumerable</span></a>.&nbsp;</li><li>So, we build an algorithm A' that takes n as input, computes the n<sup>th</sup> pair (i,j) and runs algorithm A on input i for exactly j steps.&nbsp;</li><li>If the algorithm A has returned a result after j (or fewer) steps on input i, then A' stops and returns i. Otherwise, A' stops and returns some canonical element of W. &nbsp;</li><li>So, as n increases, all pairs are tested, and algorithm A' enumerates W (repetitions are OK).</li></ul><div><br /></div><div>So, thanks to Theorem 1, we know that each effectively enumerable set is also the numerical domain of some algorithm.</div><div><br /></div><div>Since an algorithm is a finite sequence of characters in some formal language, the set of algorithms is effectively enumerable. Therefore, the set of numerical domains is also effectively enumerable. We can thus speak of the i<sup>th</sup> numerical domain W<sub>i</sub>, which is also the i<sup>th</sup> effectively enumerable set.</div><div><br /></div><div>Now let's define the set K of natural numbers as follows:<br /><br /><div style="text-align: center;">K = { i | i ∈&nbsp;W<sub>i&nbsp;</sub>}</div><br />While K is effectively enumerable (as can be proved in a way similar to the proof sketch of the "if" direction of Theorem 1 above), its complement is not!<br /><ul></ul><blockquote class="tr_bq"><b>Theorem 2:&nbsp;</b><span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>&nbsp;is not effectively enumerable.</blockquote></div><div>Proof sketch using Cantor's diagonalization&nbsp;method:<br /><ul><li>By definition, <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;"> K</span>&nbsp;= { i | i ∉&nbsp;W<sub>i&nbsp;</sub>}.</li><li>If 0&nbsp;∈&nbsp;W<sub>0</sub>&nbsp;then&nbsp;0&nbsp;∈ K, thus&nbsp;0&nbsp;∉&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>. Conversely, if 0&nbsp;∉&nbsp;W<sub>0</sub>&nbsp;then&nbsp;0&nbsp;∉&nbsp;K, thus&nbsp;0&nbsp;∈&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>. In all cases, <span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>&nbsp;≠ W<sub>0</sub>.</li><li>The same reasoning holds for&nbsp;W<sub>1</sub>,&nbsp;W<sub>2</sub>,&nbsp;W<sub>3</sub>, etc.</li><li>Therefore&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>&nbsp;is not equal to any of the effectively enumerable sets. &nbsp;</li></ul><br />So,&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">K</span>&nbsp;is our first example of a set that is enumerable (it must be, since every subset of ℕ is enumerable) but not effectively so.<br /><br />In conclusion, the set of "effectively enumerable sets" is exactly the same as the set of "numerical domains of algorithms" (by theorem 1) and we have exhibited one set that is enumerable but is not effectively enumerable.</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/OAfg--pFlwI" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/an-enumerable-but-not-effectively.htmltag:blogger.com,1999:blog-6792710671733445593.post-6054352150533537742013-06-22T19:57:00.001-07:002013-06-22T19:57:54.985-07:00Capturing numerical properties in a formal language of arithmetic<a href="http://www.amazon.com/Introduction-G-f6dels-Cambridge-Introductions-Philosophy/dp/0521674530"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> starts Chapter 4 by describing&nbsp;<i>L<sub>A</sub></i>, a formal language that is at the core of several AFTs (<a href="http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.html"><span class="Apple-style-span" style="color: blue;">axiomatized formal theories</span></a>) of arithmetic. Then Smith explains what it means for a formal language of arithmetic to "express" a numerical property and the stronger notion of "capturing" a numerical property.<br /><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Here is the definition of&nbsp;<i>L<sub>A</sub></i>:<br /><a name='more'></a></div></div><div><ul><li><b>Logical vocabulary</b>: the symbols for variables, the usual logical connectives and quantifiers, the identity (or equality) symbol, and parentheses.</li><li><b>Non-logical vocabulary</b>: the symbol for the constant 0, the unary successor function symbol S, and the binary function symbols&nbsp;+ and ×.</li><ul><ul></ul></ul><li>The <b>numerals</b> are 0, S0, SS0, SSS0, etc., which we will sometimes abbreviate 0, 1, 2, 3, etc., respectively.&nbsp;<span style="border-top: 2px solid;"><b>n</b></span>&nbsp;represents the numeral for the integer value that n stands for. So, if n stands for 5, then&nbsp;<span style="border-top: 2px solid;">n</span>&nbsp;represents SSSSS0.</li><li><b>Terms</b>&nbsp;are numerical expressions. The set of terms includes 0 and any variable, as well as any expression built up from those using the functions S,&nbsp;+ and ×. So, SS0&nbsp;× y and&nbsp;S(x + S<span style="border-top: 2px solid;">n</span>) are examples of terms. <b>Closed terms</b> are variable free.</li><li>An <b>atomic well-formed formula</b> (or atomic <b>wff</b>) has the form&nbsp;σ = τ, where&nbsp;σ and τ&nbsp;are terms. Note that = is the only predicate in&nbsp;<i>L<sub>A</sub></i>.</li><li>The wff's are formed from atomic wff's using the connectives and quantifiers, as usual.</li></ul>The interpretation<i> I<sub>A</sub></i>&nbsp;assigns the usual meanings to the non-logical vocabulary of&nbsp;<i>L<sub>A</sub></i>.</div><div><br /></div><div>The language&nbsp;<i>L<sub>A</sub></i>&nbsp;represents numerical properties using (open) wff's containing one free variable. For example, the following wff&nbsp;of&nbsp;<i>L<sub>A</sub></i>&nbsp;represents the property of being even:</div><div style="text-align: center;"><br />∃v (SS0 × v = x)</div><div><br />where x is a free variable standing in for an even number. More precisely, the statement "n is even" would be represented by the wff:&nbsp;∃v (SS0 × v =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;in which the numeral for n is substituted for all of the occurrences of x.<br /><br /><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Similarly, the language&nbsp;<i>L<sub>A</sub></i>&nbsp;represents relations between two (or more) numbers using (open) wff's containing two (or more) free variables. For example, the following wff&nbsp;of&nbsp;<i>L<sub>A</sub></i>&nbsp;represents the relation between x and y stated as "y is a multiple of x:"</div></div><div style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">∃v (x × v = y)</div></div><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">where x and y are free variables. More precisely, the statement "m is a multiple of n" would be represented by the wff:&nbsp;∃v (<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>&nbsp;×&nbsp;v =&nbsp;<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>).</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">For a wff&nbsp;of&nbsp;<i>L<sub>A</sub></i>&nbsp;to represent a property, the set of numbers that have this property must make this wff true according to&nbsp;<i>I<sub>A</sub></i>. More precisely, we say that a <b>property P is<i> expressed</i> by the open wff&nbsp;φ(x) </b>if and only if, for every n:<br /><ul><li>If n has property P, then φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true according to<i>&nbsp;I<sub>A</sub></i>.</li><li>If n does not have property P, then ¬ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true according to<i>&nbsp;I<sub>A</sub></i>.</li></ul><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Similarly, we say that <b>a relation R is <i>expressed</i> by the open wff&nbsp;φ(x,y)</b> if and only if, for every m and n:</div><ul><li>If m has the relation R to n, then φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true according to<i>&nbsp;I<sub>A</sub></i>.</li><li>If m does not have the relation R to n, then&nbsp;¬&nbsp;φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is true according to<i>&nbsp;I<sub>A</sub></i>.</li></ul><div><br />The fact that an AFT of arithmetic can express a numerical property or relation only depends on the expressive power of its interpreted formal language. It is independent of its proof system. Since we care very much about proofs, we do not only need for wff's to express numerical properties, we would also like for the AFT to prove these wff's. This is what the "capture" concept is about.</div></div><div><br /></div><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">We say that an&nbsp;AFT<b> T&nbsp;<i>captures </i>the property P&nbsp;by the open wff&nbsp;φ(x)&nbsp;</b>if and only if, for every n:</div><ul><li>If n has property P, then T ⊢ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)<i>.</i></li><li>If n does not have property P, then T ⊢ ¬ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>).</li></ul><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Similarly, we say that an&nbsp;AFT<b> T&nbsp;<i>captures&nbsp;</i>the relation R by the open wff&nbsp;φ(x,y)&nbsp;</b>if and only if, for every m and n:</div></div><ul><li>If m has the relation R to n,&nbsp;then T ⊢ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>).</li><li>If m does not have the relation R to n, then T ⊢ ¬ φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">m</span>,<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>).</li></ul><div><br /></div></div></div><div>Smith emphasizes that the way he uses the words "express" and "capture" is his own, but that there is no standard terminology in this area anyway. He likes the word 'capture' for the stronger property because 'CAPture' is a good mnemonic for 'CAse-by-case Proof.' I think what he means here is that, in the definition for 'capturing P' above, each&nbsp;φ(<span style="border-top-color: initial; border-top-style: solid; border-top-width: 2px;">n</span>)&nbsp;is proved separately for each value of n, which is very different from proving a single, universally quantified wff that would represent the entire set of numbers with property P.</div><div><br /></div><div>In conclusion:</div><div><ul><li>Capturing a property (or relation) is stronger than expressing it: A language may be expressive enough to express a property (or relation) but its axioms and inference rules may not be strong enough to capture it, that is, to prove it (even case by case).</li><li>However, if T is a <i>sound</i> AFT and T captures P with the wff&nbsp;φ(x), then&nbsp;φ(x)&nbsp;does express P, since, in a sound theory, every theorem is true in its own interpretation.</li></ul></div></div></div></div><div><ul><ul><ul></ul></ul></ul></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/koatRMtcOQo" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/capturing-numerical-properties-in.htmltag:blogger.com,1999:blog-6792710671733445593.post-89853631516321668232013-06-21T15:32:00.001-07:002013-06-21T15:32:44.715-07:00Formal systems or axiomatized formal theoriesIn chapter 3, <a href="http://www.amazon.com/Introduction-G-f6dels-Cambridge-Introductions-Philosophy/dp/0521674530"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> defines formal systems or, as he calls them, <b>axiomatized formal theories</b>&nbsp;(I will use <i>AFT</i> as an abbreviation for this phrase).<br /><br />A theory T is an <b>AFT</b> if...<br /><a name='more'></a><ol><li>T is couched in an interpreted formal language, that is, a formal language L of well-formed formulas (<b>wff</b>'s) that are meaningless sequences of symbols, together with an interpretation I that gives a meaning to the symbols and wff's in L, such that:</li><ul><li>The property of being a wff in L is <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively decidable</span></a>.</li><li>The function that computes the truth values of wff's is <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively computable</span></a>.</li></ul><li>T includes axioms such that whether or not a wff is an axiom is an effectively decidable property.</li><li>T includes a proof system (or deductive system or inference rules) such that whether or not a sequence of wff's is a valid sequence of applications of inference rules is&nbsp;an effectively decidable property.</li><li>It is effectively decidable whether or not a sequence of wff's is a proof in T, that is, a valid sequence of applications of inference rules starting from T's axioms.&nbsp;</li></ol><div>A few comments about the interpreted language of T:&nbsp;</div><div><ul><li>the language L is made up of a finite alphabet of symbols and syntactic rules for building wff's out of alphabet symbols.&nbsp;</li><ul><li>The alphabet contains two types of symbols:</li><ul><li>The <b>logical vocabulary</b> contains the symbols for variables, logical connectives&nbsp;(we assume from now on that the logical vocabulary of L contains the negation connective ¬), quantifiers, the identity (or equality) symbol, and parentheses.</li><li>The <b>non-logical vocabulary</b> contains the symbols for constants (e.g., the constant 0), predicates (e.g., the predicate/property of being even) and functions (e.g., the successor function).</li></ul><li>The syntactic rules of L define the "grammar" of the language. For example,&nbsp;¬ is a unary prefix operator while ∨ and ∧ are binary infix operators.</li></ul><li>The interpretation contains a domain D of objects and a mapping from:</li><ul><ul><li>the constants of L to D, giving meaning to the constants</li><li>the predicate symbols of L to properties and relations in D, giving meaning to the predicates</li><li>the function symbols of L to functions over D, giving meaning to the functions</li></ul></ul></ul><div><br /></div>The middle section of Chapter 3 defines important notation and concepts, some of which we have discussed previously:</div><div><ol><li>The notation <b>T&nbsp;⊦ w</b> means that w is a <b>theorem</b> of T, that is, T proves w.</li><li>T is <b>sound</b> if every theorem of T is true according to I.</li><li>T is <b>decidable</b> if the property of being a theorem of T is decidable.&nbsp;Note that decidability (i.e., always being able to tell if any given wff is a theorem) is a stronger property than being able to tell if a given sequence of wff's is a valid proof (see part 4 of the definition of AFT above). The difference is akin to the difference between being able to discover a proof and being able to check a known proof.</li><li><b>T decides w</b> if either&nbsp;T&nbsp;⊦ w or&nbsp;T&nbsp;⊦&nbsp;¬w.&nbsp;</li><li><b>T correctly decides w</b>&nbsp;if T&nbsp;⊦ w whenever w is true and &nbsp;T&nbsp;⊦&nbsp;¬w whenever w is false according to I.</li><li>T is <b>negation-complete</b> if T decides every <b>sentence</b> (i.e., wff with no <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.html"><span class="Apple-style-span" style="color: blue;">free variables</span></a>) in L.&nbsp;Recall the&nbsp;<a href="http://summerofgodel.blogspot.com/2013/06/peter-smiths-overview-of-godels.html"><span class="Apple-style-span" style="color: blue;">distinction between negation completeness and semantic completeness</span></a>.</li><li>T is <b>consistent</b> if there is no sentence w such that&nbsp;T&nbsp;⊦ w and T&nbsp;⊦&nbsp;¬w.</li></ol><div>Let's emphasize the difference between decidability and negation-completeness.</div><div><ul><li>An AFT is negation-complete if its proof system and axioms are powerful enough to prove or disprove any wff of its language.</li><li>An AFT is decidable if there exists an algorithm that can determine whether any wff is one of its theorems.</li></ul><div>So, while all negation-complete AFTs are decidable (see below), a decidable&nbsp;AFT does not have to be negation-complete. For example, a propositional AFT (i.e., without quantifiers) is decidable, since we can use truth tables to determine which propositions follow from the axioms, but the axioms may not be enough to prove or disprove all wff's. For example, if the logical language includes the propositional variables p and q and the AFT contains the single axiom&nbsp;¬p (that is, p is false), then the AFT cannot prove nor disprove the wff p&nbsp;∨ q (because T cannot infer the truth value of q). Nevertheless, this AFT is trivially decidable (e.g.,&nbsp;a truth table tells us that&nbsp;p&nbsp;∨ q is not a theorem).</div></div></div><div><br />The last part of Chapter 3 proves important properties of AFTs, namely:<br /><ol><li>The set of wff's of any AFT is <a href="http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.html"><span class="Apple-style-span" style="color: blue;">effectively enumerable</span></a>.</li><li>The set of sentences of any AFT is&nbsp;effectively enumerable.</li><li>The set of proofs of any AFT is&nbsp;effectively enumerable.</li><li>The set of theorems of any AFT is&nbsp;effectively enumerable.</li></ol><div>In general, <i>effective decidability is a stronger property than effective enumerability</i>.&nbsp;Being able to enumerate the theorems of an AFT does NOT imply that the AFT is decidable. Even if we are able to list all of the theorems, determining that a wff is not a theorem may require going through the whole list of theorems, which is often infinite. Therefore, this approach is not an algorithm: it does not always terminate in a finite number of steps.&nbsp;However...</div><div><ol start="5"><li>All negation-complete theories <i>are</i> decidable.</li></ol><div>There are two cases to consider in a proof sketch.&nbsp;If the AFT is consistent, given any wff w, one can simply go through the enumeration of its theorems and stop as soon as either w or&nbsp;¬w is found (which is guaranteed to happen since the AFT is negation-complete). If the AFT is inconsistent, then its set of theorems is identical to its set of wff's (i.e., in this case, <a href="http://summerofgodel.blogspot.com/2013/05/absolute-proof-of-consistency-of-fsn.html"><span class="Apple-style-span" style="color: blue;">the AFT proves all of its wff's</span></a>); so the AFT is, by definition, decidable.</div></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/andpVhGyCho" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/formal-systems-or-axiomatized-formal.htmltag:blogger.com,1999:blog-6792710671733445593.post-54282497620477445392013-06-19T19:15:00.001-07:002013-06-19T19:19:51.278-07:00Effective computability, decidability and enumerabilityIn Chapter 2, Smith covers familiar ground. So this post will be short. Here is a quick summary, section by section.<br /><br />Section 1 reviews terminology pertaining to functions, namely the notions of domain and range, as well as special cases of functions, namely injective (or one-to-one) functions, surjective (or onto) functions and bijective functions (or one-to-one correspondences).<br /><a name='more'></a><br />Section 2 defines effective computation.&nbsp;A function is <b>effectively computable</b> if there exists a mechanical procedure (or algorithm) to compute the value of the function for every element in its domain.<br /><br />A property is <b>effectively decidable</b>&nbsp;if there exists a mechanical procedure (or algorithm) that, given any element, will determine whether or not the element has the property.&nbsp;Similarly, an n-ary relation is&nbsp;<b>effectively decidable</b>&nbsp;if there exists a mechanical procedure (or algorithm) that, given any n-tuple, will determine whether or not the relation holds among the n elements of the tuple.<br /><br /><div>To get a sense of what an algorithm is (since it is not a concept that can be defined mathematically), Smith briefly discusses the <a href="http://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis"><span class="Apple-style-span" style="color: blue;">Church-Turing thesis</span></a>, which defines algorithms as&nbsp;<a href="http://en.wikipedia.org/wiki/Turing_machine"><span class="Apple-style-span" style="color: blue;">Turing machines</span></a>.</div><div><br /></div><div>Section 3 defines enumerable sets. A set S is <b>enumerable</b> if either S is empty or there exists a surjective function from the set of natural numbers&nbsp;ℕ (including zero) to S. Equivalently, the elements of an enumerable set can be listed (or enumerated) in some order, as in: element number 0, element number 1, element number 2, etc., in such a way that each element in the set is indexed by at least one integer.</div><div><br /></div><div>Clearly,&nbsp;ℕ&nbsp;is enumerable (using the identity function, which is bijective). Similarly, any subset of&nbsp;ℕ is enumerable. For example, the set E of even natural numbers is enumerable, using the function f<sub>1</sub>:&nbsp;ℕ&nbsp;→&nbsp;E with&nbsp;f<sub>1</sub>(n) = 2n. As another example, the finite set S = {1,2,3} is&nbsp;enumerable, using the function f<sub>2</sub>:&nbsp;<span class="Apple-style-span" style="font-family: 'Arial Unicode MS', 'Lucida Grande', sans-serif;">ℕ&nbsp;</span>→ S with&nbsp;f<sub>2</sub>(0) = 1,&nbsp;f<sub>2</sub>(1) = 2,&nbsp;f<sub>2</sub>(2) = 3,&nbsp;f<sub>2</sub>(3) = 3,&nbsp;f<sub>2</sub>(4) = 3, etc.&nbsp;Note that the function does not have to be injective (that is, repetitions are allowed in the enumeration).<br /><br />Then Smith uses Cantor's classic<span class="Apple-style-span" style="color: blue;"> <a href="http://en.wikipedia.org/wiki/Cantor's_diagonal_argument"><span class="Apple-style-span" style="color: blue;">diagonalization proof technique</span></a></span> to prove that the following sets are&nbsp;<b>indenumerable</b>, that is, NOT enumerable:<br /><ul><li>the set of infinitely-long bit strings</li><li>the set of real numbers between 0 and 1</li><li>the set of subsets of&nbsp;<span class="Apple-style-span" style="font-family: 'Arial Unicode MS', 'Lucida Grande', sans-serif;">ℕ</span></li><li>the set of functions from ℕ&nbsp;→&nbsp;{0,1}</li></ul><div><br />Section 4 defines effective enumerability.&nbsp;A set S is <b>effectively enumerable</b>&nbsp;if either S is empty or there exists an effectively computable surjective function from the set of natural numbers&nbsp;ℕ&nbsp;(including zero) to S.</div></div><div><br /></div><div>So, according to this taxonomy, a set can be:</div><div><ol><li>effectively enumerable (e.g., the empty set, all finite sets,&nbsp;<span class="Apple-style-span" style="font-family: 'Arial Unicode MS', 'Lucida Grande', sans-serif;">ℕ</span>), or</li><li>enumerable but not effectively so (no example yet), or</li><li>not enumerable (e.g.,&nbsp;<span class="Apple-style-span" style="font-family: 'Arial Unicode MS', 'Lucida Grande', sans-serif; font-size: 14px;">ℝ)</span>.</li></ol><div>Section 5 concludes the chapter by going over the well-known enumeration of the set of pairs of natural numbers by traversing the 2D table of pairs in a zigzag pattern along the anti-diagonals, thereby proving that&nbsp;ℕ<sup>2</sup>&nbsp;is effectively enumerable.</div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/9OJF2pp4b1w" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/effective-computability-decidability.htmltag:blogger.com,1999:blog-6792710671733445593.post-8216606138097087382013-06-18T19:00:00.000-07:002013-06-18T19:00:00.761-07:00Peter Smith's overview of Gödel's incompleteness theoremsFinally! As was my initial goal, I now turn my attention to <a href="http://www.amazon.com/Introduction-G-f6dels-Cambridge-Introductions-Philosophy/dp/0521674530"><span class="Apple-style-span" style="color: blue;">Peter Smith's book</span></a> on the proof of Gödel's incompleteness theorems (by the way, I own the first edition of his book; so that is the one I will be referring to). I did not expect to spend that much time on Nagel and Newman's book. But it was worthwhile in getting the big picture.<br /><br />Chapter 1 is an introductory and user-friendly discussion of topics that were mentioned in earlier posts, such as basic arithmetic, formal systems, (in)completeness, consistency, the statement of Gödel's incompleteness theorems and some of their implications. Therefore, in this post, I will just highlight the points where Smith provides new information or has a different perspective.<br /><a name='more'></a><ol><li>Smith makes the distinction between <i>proving</i> a theorem and <i>deriving</i> a theorem in a formal system. While "prove" and "derive" can be seen as synonyms, "proved" is often interpreted as "proved to be true," whereas "derived in a formal system" just means "resulting from the application of inference rules starting from axioms" without any assumption that the theorems of the system are true. If the axioms of the system are all true (in some interpretation) and its rules of inference are truth-preserving, then the theorems of the system are all true. In this case, we say that the system is <b>sound</b>. In general, however, soundness is not a required feature of formal systems. So the term "derive" is more neutral with respect to truth than the term "prove."</li><li>Smith makes another, more important distinction between two types of completeness, also related to the concept of truth. He defines <b>negation completeness</b>, which is different from <b>semantic completeness</b> that we discussed <a href="http://summerofgodel.blogspot.com/2013/06/proof-sketch-of-godels-incompleteness.html"><span class="Apple-style-span" style="color: blue;">here</span></a>.&nbsp;A formal system is negation complete if, for every well-formed formula (or wff) w in the system, either w is a theorem or ¬ w is a theorem. In other words, every wff of the formal system is <b>decidable</b> (either it or its negation is a theorem). Notice that negation-completeness is independent of the truth of the theorems in the system. Each and every theorem in a negation-complete system could be false and every truth may be a non-theorem in this system. In contrast, Nagel and Newman framed the first incompleteness theorem in terms of semantic completeness, whereby all true statements are theorems. While these two types of completeness can be related to each other through the concept of soundness, it appears to be worthwhile to keep the distinction in mind.</li><li>Smith presents a slightly different perspective on the first theorem and the "incompletability" of formal systems of arithmetic. <a href="http://summerofgodel.blogspot.com/2013/06/proof-sketch-of-godels-incompleteness.html"><span class="Apple-style-span" style="color: blue;">Recall</span></a>&nbsp;how adding Gödel's sentences (as axioms) to a formal system cannot make it complete, since a Gödel sentence can always be constructed for such an augmented formal system. Repeating this process of augmentation will yield an infinite number of increasingly strong (yet still incomplete) formal systems and associated Gödel sentences. None of these Gödel sentences can be derived in the original formal system. Therefore, there is an infinite number of true Gödel sentences that the original formal system cannot derive.</li><li>Smith makes a good point regarding the second theorem: what would be the value of a proof of consistency of F <i>within</i> F, even if such a proof could be obtained? Recall point 1 above: finding a derivation of a theorem T in F does not mean that T is true. In fact, <a href="http://summerofgodel.blogspot.com/2013/05/absolute-proof-of-consistency-of-fsn.html"><span class="Apple-style-span" style="color: blue;">as we already argued</span></a>,&nbsp;if a formal system is inconsistent, it derives every single wff and its negation, including the wff that "states" it own consistency! In other words, a proof of this wff would not provide much support for its truth. So the fact that this proof does not exist does not appear to be such a negative result. So the impact of the second theorem must lie elsewhere: indeed, if an augmented formal system (with added axioms, see point 3 above) is consistent, then so is the initial formal system, since removing axioms cannot introduce inconsistencies. Furthermore, if a formal system could prove the consistency of an augmented version of itself, it could prove its own consistency as well. Therefore, a formal system cannot prove its own consistency nor the consistency of any richer formal system, which is a serious blow to <a href="http://summerofgodel.blogspot.com/2013/05/absolute-proofs-of-consistency-and-meta.html"><span class="Apple-style-span" style="color: blue;">Hilbert's programme</span></a>, whose goals included finding finitary proofs of consistency for non-finitary systems.&nbsp;</li><li>Smith describes how to mirror arithmetic within set theory, which implies that no formal system for set theory can be negation complete, using a reduction similar to those we discussed in <a href="http://summerofgodel.blogspot.com/2013/05/relative-proofs-of-consistency.html"><span class="Apple-style-span" style="color: blue;">relative proofs of consistency</span></a>.</li></ol><br /><br /><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/B6MJkLUITLA" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/peter-smiths-overview-of-godels.htmltag:blogger.com,1999:blog-6792710671733445593.post-71475957014562881962013-06-17T18:05:00.001-07:002013-06-27T19:43:41.447-07:00Proof sketch of Gödel's incompleteness theoremsIn this post, I will follow the outline of the proof given in Section VII of Nagel and Newman's book.&nbsp;Recall that:<br /><div><ul><li>a formal system F is <b>consistent</b> if there is no well-formed formula (wff) w such that F proves both w and ¬ w, and</li><li>a formal system is (semantically)&nbsp;<b>complete</b> if it can prove all of the true wff's that it can express.</li></ul><div>Now, Gödel's first incompleteness theorem can be paraphrased as:<br /><blockquote class="tr_bq">Any consistent formal system that is expressive enough to model arithmetic is both incomplete and "incompletable."</blockquote><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">and&nbsp;Gödel's second incompleteness theorem can be paraphrased as:</div><blockquote class="tr_bq"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Any consistent formal system that is expressive enough to model arithmetic cannot prove its own consistency.</div></blockquote>According to Nagel and Newman, a proof of the first theorem can be sketched in 4 steps:<br /><a name='more'></a><br /><ul><li>Step 1: Build a wff (call it G) in the formal system F such that the meta-mathematical interpretation of G is: "G is not provable in F."</li><li>Step 2: Show that: "G is provable in F" if and only if "<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬&nbsp;</span></span>G is provable in F." In other words, either both G and&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬&nbsp;</span></span>G are provable in F or neither one of them is. Therefore, if F is consistent, then neither G nor&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ </span>G is provable in F.</li><li>Step 3: Show (using meta-mathematical reasoning) that G is true. Therefore, G is true but not provable in F, hence...</li><li>Step 4: F is incomplete. Also, adding G as an axiom to F would make G (trivially) provable in the new formal system F'. However, then we could build another wff, say G', that is both true but unprovable in F'. This process can be repeated forever, meaning that it is not possible to make F complete simply by adding Gödel's sentences as axioms. In short, any consistent and sufficiently expressive formal system is "incompletable."</li></ul><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Now, here is Nagel and Newman's proof sketch of Gödel's second theorem:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><ul><li>Step 1: Build a wff (call it C) of the formal system F such that the meta-mathematical interpretation of C is: "F is consistent."</li><li>Step 2: Prove the wff C → G in F (recall that C → G means "C implies G" and is logically equivalent to ¬ C ∨ G).</li><li>Step 3: Sub-proof by contradiction:</li><ul><li>Assume C is provable in F.</li><li>By modus ponens applied to the premises C and&nbsp;C → G, we can prove G in F.</li><li>But G is not provable in F; so we have exhibited a contradiction.</li><li>Therefore, C is not provable in F.</li></ul></ul><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />The crucial step in&nbsp;Gödel's proofs is a constructive one: he showed how to build in F a wff G with&nbsp;Gödel number g whose meta-mathematical interpretation is the following: "The wff with&nbsp;Gödel number g is not provable in F." In other words, the wff G can be interpreted (at the meta-level) to mean that G is not provable within the formal system. In short, the meta-mathematical interpretation of G is "I am not provable."</div><br />Therefore, the crux of the proofs is how to build Gödel's sentence, that is, the wff G (step 1 of the first proof).<br /><br />Recall the following expressions in F (on the right) and their meta-mathematical interpretations (on the left) that we described in <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-arithmetization-of-meta.html"><span class="Apple-style-span" style="color: blue;">this post</span></a>:<br /><table border="0"><tbody><tr><td style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><table border="0"><tbody><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><th><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">meta-mathematical statement</div></th><th><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">⇔</div></th><th><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">mathematical proposition in F</div></th></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><td style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">the sequence s is a proof of w</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">⇔</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></td><td style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>ProofOf( Φ(s) , Φ(w) )</i></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></td></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><th>meta-mathematical numerical expression </th><th>⇔ </th><th>mathematical/numerical expression in F </th></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><td style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">the G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number of the wff</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">obtained by substituting (the numeral in F for) the number n for the free variable x</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">in the wff with Gödel number m</div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">⇔</div></td><td style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub( n , 17 , m )</i></span></i></div></td></tr></tbody></table></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left;">where&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">Φ(w) is the&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">Gödel number of the wff w and&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">Φ(s) is the&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">Gödel number of s, which is a sequence of wff's.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><br /></span>Now, to the construction of Gödel's sentence G...</div><blockquote class="tr_bq"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">The wff&nbsp;</span><i>ProofOf(y,z)</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">&nbsp;means "the sequence with </span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number y&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">is a proof of the wff with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number z."</blockquote><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left;">Hence:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left;"></div><blockquote class="tr_bq"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">The wff &nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span><i>P</i><i>roofOf(y,z)</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">&nbsp;means "the sequence with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number y&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">is NOT a proof of the wff with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number z."</blockquote><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: left;">Therefore:</div><blockquote class="tr_bq"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">The wff <i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf(y,z)</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>)</i>&nbsp;means "For every integer y, the sequence with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number y&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">is NOT a proof of the wff with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number z."</blockquote><div style="text-align: left;">In other words:&nbsp;</div><blockquote class="tr_bq"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">The wff&nbsp;<i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf(y,z)</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>)</i>&nbsp;means "T</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">he wff with&nbsp;</span>G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number z is not provable in F."</blockquote><div style="text-align: left;">Note that&nbsp;z is a free variable&nbsp;in the wff&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf(y,z)</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>)</i><i>. </i></span>So we'll refer to this wff as <b>G(z)</b>.&nbsp;</div><div style="text-align: left;"><br /></div><div style="text-align: left;">G(z) is neither true nor false until we assign a specific numerical value to z. If we could find a value v such that G(v) has Gödel number v, then we would be done!&nbsp;Recall that G(v) means "the wff with Gödel number v is not provable."&nbsp;So, if&nbsp;Φ(G(v)) = v, then G(v) means "I am not provable" and G(v) is Gödel's sentence G.&nbsp;</div><div style="text-align: left;"><br /></div><div style="text-align: left;">The last step is to find this value v.&nbsp;</div><div style="text-align: left;"><br /></div><div style="text-align: left;">It is now time to use the second formal expression in F&nbsp;described above, namely:<br /><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i><br /></i></span></i><br /><div style="text-align: center;"><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub( n , 17 , m</i></span></i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>&nbsp;)</i></span></i></div><br />In fact, let's consider, as a special case of it:<br /><br /><div style="text-align: center;"><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub( x , 17 , x</i></span></i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>&nbsp;)</i></span></i></div><br />which represents in F the Gödel number α of the wff obtained by substituting the number x for the free variable x in the wff with Gödel number x.</div><div style="text-align: left;"><br />Since&nbsp;α is a number, we can replace it (that is, its numeral in F) for the numerical variable z in G(z), yielding:<br /><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf(y,&nbsp;</i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(x,17,x</i></span></i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>) ))</i></span></i></div><i><br /></i></div><div style="text-align: left;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">Obviously, this wff has a G</span>ödel number. Let's call this number h and see what happens when we replace the free variable x above by the numeral for h:<br /><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf( y ,</i>&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>,17,h)&nbsp;</i></span></i><i>)&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>)</i></span></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i><br /></i></span></div>It turns out that this wff is G! Why? Because...<br /><ul><li>this wff means&nbsp;"the wff with Gödel number&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i style="font-style: italic;">,17,h)</i> is not provable."&nbsp;</span></li><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">this&nbsp;</span>wff is the wff with Gödel number&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i style="font-style: italic;">,17,h)</i></span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">remember how we built this wff, namely by&nbsp;substituting h for x in&nbsp;the wff with Gödel number h.</span></li></ul></ul><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">Therefore, the value of v we were looking for is&nbsp;</span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i style="font-style: italic;">,17,h).&nbsp;</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;">In conclusion, G</span>ödel's sentence G is G(v) = G(&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i style="font-style: italic;">,17,h)</i></span>&nbsp;), that is:<br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i><br /></i></span> <br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>∀y (</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">¬</span>ProofOf( y ,</i>&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub(h</i></span></i><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>,17,h)&nbsp;</i></span></i><i>)&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>)</i></span></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div></td><td style="text-align: center;"><br /></td></tr></tbody></table></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/zkytcRGBy-k" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/proof-sketch-of-godels-incompleteness.htmltag:blogger.com,1999:blog-6792710671733445593.post-74687340615872786352013-06-07T22:07:00.000-07:002013-06-07T22:07:52.495-07:00Mappings and the arithmetization of meta-mathematicsIn an <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.html"><span class="Apple-style-span" style="color: blue;">earlier post</span></a>, we discussed mappings in general and then described Gödel numbering, which is a mapping between the set of well-formed formulas (<b>wff</b>'s) in a formal system and the set of positive integers.<br /><br />In part B of section VII, Nagel and Newman describe a more interesting mapping used in Gödel's proof. In this mapping, the domain is&nbsp;the set of meta-mathematical statements about structural properties of wff's, while the co-domain is the set of wff's. This post describes how this mapping enabled&nbsp;Gödel to arithmetize meta-mathematics.<br /><a name='more'></a><br />Let's use the letter F to refer to the formal system used in Gödel's proof. Recall that, thanks to <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.html"><span class="Apple-style-span" style="color: blue;">Gödel numbering</span></a>, each wff in F maps to a positive integer. Therefore, each meta-level statement about wff's&nbsp;and their relationships is also a statement about positive integers (Gödel numbers) and their relationships.<br /><br />As a simple example, let's consider the following definition:<br /><blockquote class="tr_bq"><i>A <b>universally quantified</b> wff is a wff of F that starts with the universal quantifier, that is, a wff of the form ∀x (...) or&nbsp;</i><i>∀y (...) or ...</i></blockquote>This definition is part of meta-mathematics because it talks about a structural property of wff's of F, that is, how they are built out of the alphabet symbols. So, any wff of the form&nbsp;∀x (...)&nbsp;with the ellipse filled in, is part of mathematics (it is a wff inside F), while the definition above is part of meta-mathematics (it is outside of F and about some of the wff's of F).<br /><br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">For instance, <a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.html"><span class="Apple-style-span" style="color: blue;">recall</span></a> the wff:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∀x&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;= f(x) ) )</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">This wff is a universally quantified wff since it satisfies the definition given above. The G</span>ödel number of this wff is:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="text-align: center;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">2<sup>9</sup>&nbsp;×&nbsp;3<sup>17</sup>&nbsp;×&nbsp;5<sup>11</sup></span>&nbsp;× 7<sup>5</sup>&nbsp;× 11<sup>11</sup>&nbsp;× 13<sup>17</sup>&nbsp;× 17<sup>15</sup>&nbsp;× 19<sup>3</sup>&nbsp;× 23<sup>11</sup>&nbsp;× 29<sup>17</sup>&nbsp;× 31<sup>13</sup>&nbsp;×&nbsp;37<sup>13</sup>&nbsp;× 41<sup>13</sup></div><div><sup><br /></sup></div><div style="text-align: left;">What is common between this Gödel number and the Gödel number of all other universally quantified wff's?<br /><br />Since the first prime in each Gödel number is always 2 and its power is the Gödel number of the first symbol in the wff (namely 9), all universally quantified wff's have&nbsp;2<sup>9</sup>&nbsp;(= 512) as their first factor.<br /><br />Moreover, all of the remaining terms in the product (i.e., the Gödel number) are primes larger than 2, and thus odd numbers. Therefore, we just showed the following equivalence:<br /><br /><div style="text-align: center;">(a) a wff w in F is universally quantified</div><div style="text-align: center;"><br /></div><div style="text-align: center;">if and only if</div><div style="text-align: center;"><br /></div><div style="text-align: center;">(b) the Gödel number of w is divisible by 512&nbsp;</div><div style="text-align: center;">but is not divisible by any even integer larger than 512</div><br />The next step is to realize that the numerical properties of being "divisible by some integer," of being "even," and of being "larger than 512" are all representable by wff's in F.<br /><br />In conclusion, statement (b) above can be represented by some wff in F. So, the meta-mathematical statement (a) is true if and only if the wff expressing the mathematical statement (b) is true under the intended interpretation of F.<br /><br />More generally, Gödel proved that EVERY meta-mathematical statement about the structural properties of wff's can be mapped to some wff in F. This is the so-called <b>arithmetization of meta-mathematics</b>, that is, a mapping from the set of meta-mathematical statements to the set of mathematical (and more precisely arithmetical) statements in the formal system F.<br /><br />This mapping is important because it means that proving any meta-mathematical statement about the structural properties of formulas can be reduced to proving wff's within the formal system F. In other words, this mapping reduces meta-level reasoning to reasoning within the formal system.<br /><br />So, coming back to our example, proving that some formula w is universally quantified reduces to proving in F the wff&nbsp;that represents the statement (b) above.<br /><br />Furthermore, this mapping established by Gödel is not limited to reducing structural <i>properties of wff's</i> to numerical properties (of Gödel numbers). It extends to <i>relationships among two or more wff's</i>.<br /><br />So&nbsp;Gödel proved that EVERY meta-mathematical statement about structural relationships between two wff's w<sub>1</sub> and w<sub>2</sub> can be mapped to some wff in F, and that this wff represents a specific (numerical) relationship between the Gödel numbers of&nbsp;w<sub>1</sub>&nbsp;and&nbsp;w<sub>2</sub>.<br /><br />Let's define some notation that will make the rest of this post easier to follow.<br /><ul><li>If w is any wff (or sequence of wff's) in F, let <b>Φ(w)</b>&nbsp;denote the&nbsp;Gödel number of&nbsp;w.&nbsp;</li><li>If&nbsp;w<sub>1<i style="font-weight: bold;">&nbsp;</i></sub>and&nbsp;w<sub>2</sub>&nbsp;are two wff's in F, let&nbsp;<b>Prefix(w<sub>1</sub>,w<sub>2</sub>)</b> denote the meta-mathematical statement "the wff w<sub>1</sub>&nbsp;is a prefix of the wff&nbsp;w<sub>2</sub>," that is, the&nbsp;wff w<sub>1</sub>&nbsp;not only appears inside (as a substring of) wff&nbsp;w<sub>2</sub>, but it appears right at the beginning of&nbsp;w<sub>2</sub>. So for example:</li></ul><div style="text-align: center;">Prefix( (0=0) , (0=0) ∨ (f(0)=0) ) is true,</div><div style="text-align: center;"><br /></div><div style="text-align: center;">but</div><div style="text-align: center;"><br /></div><div style="text-align: center;">Prefix( (0=0) ,&nbsp;(f(0)=0)&nbsp;∨&nbsp;(0=0) ) is false.<br /><br /><div style="text-align: left;">Keep in mind that&nbsp;Prefix(w<sub>1</sub>,w<sub>2</sub>)&nbsp;is a meta-mathematical statement. It states a structural relationship between two wff's in F. But according to Gödel's mapping, this meta-mathematical statement is mapped to a numerical relationship between&nbsp;Φ(w<sub>1</sub>)&nbsp;and&nbsp;Φ(w<sub>2</sub>). In fact, Nagel and Newman use that example because the numerical relationship is trivial.<br /><br />Recall that the Gödel number of a wff is a product of prime powers, where the primes appear in increasing order starting at 2. Therefore,&nbsp;&nbsp;Prefix(w<sub>1</sub>,w<sub>2</sub>)&nbsp;is true if and only if&nbsp;Φ(w<sub>1</sub>) evenly divides Φ(w<sub>2</sub>), or equivalently,&nbsp;Φ(w<sub>2</sub>) is a multiple of&nbsp;Φ(w<sub>1</sub>). Again, the numerical relationship of "x being a multiple of y" is easily represented within F itself.<br /><br />One fascinating property that follows from Gödel's arithmetization of meta-mathematics is the fact that the formal system F is capable to describing itself!&nbsp;More specifically, it can represent numerical relationships that, when translated into meta-mathematical statements, express facts about F.<br /><br />The examples of meta-mathematical statements and corresponding numerical relationships given above are trivial and uninteresting, except for the fact that they are simple enough that we can specify them fully in terms of easily understood arithmetical properties. But Gödel proved that <i>any</i> meta-mathematical statement about structural relationships between wff's of F and sequences of wff's of F can be expressed in F.<br /><br />Now we turn our attention to a crucial meta-mathematical statement whose corresponding numerical relationship is really complex. The good news is that we do not need to fully specify this&nbsp;numerical relationship&nbsp;here. We just need to know that it exists, as Gödel proved in his paper (and I cannot wait to see what <a href="http://summerofgodel.blogspot.com/2013/05/i-just-want-to-understand-proof-of.html"><span class="Apple-style-span" style="color: blue;">Peter Smith</span></a> has to say about this proof!).<br /><br />Since the property of completeness of a formal system has everything to do with its capabilities and limitations in terms of which wff's it can and cannot prove, let us focus on the following meta-mathematical statement:<br /><br /><div style="text-align: center;">(c) the sequence of wff's s is a proof in F of the wff w</div><div style="text-align: center;"><br /></div>Since each wff (and each sequence of wff's) in F has a unique Gödel number, we can use&nbsp;Gödel's mapping to conclude that statement (c) is equivalent to<br /><br /><br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">(d) the numbers Φ(s) and&nbsp;Φ(w) are related to each other through&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">a&nbsp;(complex but well-defined) numerical relationship&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Even though the numerical relationship mentioned in (d) is far from trivial (unlike "being divisible by"), Gödel proved that it can be represented by some wff in F. Let's denote this wff by</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">&nbsp;<b>ProofOf(Φ(s),Φ(w))</b></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">In other words:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">(c) is true&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">if and only if</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">the wff&nbsp;<i>ProofOf(Φ(s),Φ(w))</i>&nbsp;is true&nbsp;in the intended interpretation of F</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Keep in mind that (c) is a <i>meta-mathematical statement </i>about the purely structural properties of wff's (namely whether the sequence s is formal proof of w), whereas&nbsp;<i>ProofOf(Φ(s),Φ(w))</i> is a <i>mathematical statement</i> within F about the numbers&nbsp;Φ(s) and Φ(w).</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">The key property of Gödel's mapping is that these two statements have the same truth value: they are either both true or both false.&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Following Nagel and Newman's lead, let's conclude this post with one more piece of notation that we'll need later.&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><a href="http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.html"><b><span class="Apple-style-span" style="color: blue;">Recall the concept of substitution</span></b></a>, by which a numerical expression n (e.g., f(f(0)) or f(y)) replaces each and every free occurrence of a numerical variable x in some wff w. Let's call w' the wff resulting from this substitution and write:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">(e) <i>w' = substitute(n,x,w)</i></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><i><br /></i></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">to mean</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><i><br /></i></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">(e') w' is the wff obtained by substituting n&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">for each free occurrence of x in the wff w</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Note that (e') is a meta-mathematical statement that describes a purely syntactical relationship between two wff's w and w', a numerical expression n and a numerical variable x.&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">In (e'), w and w' are wff's in F. x is one of the basic signs, namely, a numerical variable in the alphabet of F (any other numerical variable can be used in place of x). Finally, n is an expression in F representing a number. For example, n could be the numerical expression f(f(f(f(f(0))))) representing the number 5. Note that n is <i>not</i> the number 5, or any number, for that matter; it is an expression that denotes a number.</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">As before, using Gödel's mapping, (e') gets mapped to a numerical relationship among the numbers&nbsp;Φ(w),&nbsp;Φ(w'), Φ(x) = 17 and the number that n denotes. Again, this numerical relationship is complex, but it exists and can be represented by a wff in F. Let's denote this wff:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><i>Φ(w') =&nbsp;</i><i><b>Sub(n,17,Φ(w))</b></i></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">whose English interpretation is:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">Φ(w') is&nbsp;the&nbsp;Gödel number of the wff obtained by substituting&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">the numerical expression n for each free occurrence of&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;">the variable with Gödel number 17 in the wff w</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><br /></div><b><span class="Apple-style-span">In conclusion, we have described G</span>ö<span class="Apple-style-span">del's arithmetization of meta-mathematics, that is, a mapping from the set of meta-mathematical statements about the structural properties of wff's in F to the set of wff's in F, where F is a formal system for arithmetic.</span></b><br /><span class="Apple-style-span"><b><br /></b></span><span class="Apple-style-span">In particular, we have highlighted the two following structural relationships:</span><br /><ul><li><b>the structural relationship between a wff w and a formal proof of w</b></li><li><b>the structural relationship between a wff w and the wff resulting from substituting in w a numerical expression for a free numerical variable</b></li></ul><div>resulting in the following two associations:</div><div><br /></div><center> <table border="0"><tbody><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><th>meta-mathematical statement</th><th>⇔</th><th>mathematical proposition in F</th></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><td style="text-align: center;">the sequence s is a proof of w<br /><br /></td><td>⇔<br /><br /></td><td style="text-align: center;"><i>ProofOf( Φ(s) , Φ(w) )</i><br /><br /></td></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><th>meta-mathematical numerical expression</th><th>⇔</th><th>mathematical/numerical variable in F</th></tr><tr><th><hr /></th><th></th><th><hr /></th></tr><tr><td style="text-align: center;">the G<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">ö</span>del&nbsp;number of the wff<br />obtained by substituting n for<br />the free variable x in the wff w</td><td>⇔</td><td style="text-align: center;"><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-style: normal;"><i>Sub( n , 17 , Φ(w) )</i></span></i></td></tr></tbody></table></center></div></div></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/gHXfKKOEXgc" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/mappings-and-arithmetization-of-meta.htmltag:blogger.com,1999:blog-6792710671733445593.post-60111159540554728452013-06-06T09:17:00.002-07:002013-06-06T09:21:28.527-07:00Mappings and Gödel numberingSection VI of Nagel and Newman's book describes mappings. A <b>mapping</b> is an operation that applies to two sets called the <b>domain</b> and the <b>co-domain</b>. More specifically, a mapping associates to each element of the domain one or more elements of the co-domain.<br /><a name='more'></a><br />I found the examples of mappings in this section underwhelming. For example, the book discusses the so-called&nbsp;<span class="Apple-style-span" style="color: blue;"><a href="http://en.wikipedia.org/wiki/Richard's_paradox" target="_blank"><span class="Apple-style-span" style="color: blue;">Richard's paradox</span></a>&nbsp;</span>which is based on a mapping between the set of non-negative integers and the set of English sentences that define arithmetical properties of integers (for example, the definition of an even integer or the definition of a prime number).<br /><br />However, as the book explains well, the reasoning leading to the "paradox" is actually fallacious, since it confuses reasoning (i.e., reasoning within the system) and meta-reasoning (i.e., reasoning about, and thus, outside the system). The system under consideration here is the set of rules for building English sentences describing arithmetical properties of integers (obviously, this system is not formal).<br /><br />So there is really no paradox in this example. Nevertheless, Gödel mentions it in his 1931 paper because the construction of the mapping and the way it is used share some similarities with the mapping used by Gödel (who, of course, did not make any logical error in his paper).<br /><br />So I think it makes more sense to jump immediately to the mapping used by&nbsp;Gödel. In this post, we will discuss only the first step in&nbsp;Gödel's construction, which is described in Part A of Section VII of Nagel and Newman's book.&nbsp;This first step is the famous "Gödel numbering" scheme by which&nbsp;Gödel establishes a mapping between the set of well-formed formulas in his formal system and the set of non-negative integers.<br /><br />Before we can describe the mapping, we must describe the formal system itself, which is a combination of&nbsp;Whitehead and Russell's&nbsp;<a href="http://en.wikipedia.org/wiki/Principia_Mathematica" target="_blank"><span class="Apple-style-span" style="color: blue;">Principia Mathematica</span></a>&nbsp;and&nbsp;<a href="https://en.wikipedia.org/wiki/Peano_axioms" target="_blank"><span class="Apple-style-span" style="color: blue;">Giuseppe Peano's axioms</span></a>. We will only give enough details about the formal system to be able to explain Gödel numbers.<br /><br />Since the formal system is supposed to model the arithmetic of non-negative integers, we will occasionally refer to this intended interpretation of the symbols and well-formed formulas. Of course, the formal system is (by definition) purely syntactical. Reasoning within it does not rely on meaning at all. Nevertheless, it will help to keep the intended model in mind.<br /><br />First, the alphabet of the formal system is made up of the symbols shown in the leftmost column of Table 1 below (the rightmost column can be ignored until we turn our attention to the mapping). The first 8 basic signs (as Gödel calls them)&nbsp;are constant symbols. Their meaning is alluded to in the table.<br /><br /><center><strong>Table 1: Basic signs (i.e., alphabet symbols)</strong><br /><table border="1"><tbody><tr><td style="text-align: center;"><strong>Basic Signs of the Formal System</strong></td> <td style="text-align: center;"><strong>Gödel Numbers of the Basic Signs</strong></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">0</td><td style="text-align: center;">the numeral for zero</td></tr></tbody></table></td> <td style="text-align: center;">1</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">f</td><td style="text-align: center;">the successor function</td></tr></tbody></table></td> <td style="text-align: center;">3</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">¬</td><td style="text-align: center;">logical negation</td></tr></tbody></table></td> <td style="text-align: center;">5</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">∨</td><td style="text-align: center;">logical disjunction</td></tr></tbody></table></td> <td style="text-align: center;">7</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">∀</td><td style="text-align: center;">"for all" quantifier</td></tr></tbody></table></td> <td style="text-align: center;">9</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">(</td><td style="text-align: center;">left parenthesis</td></tr></tbody></table></td> <td style="text-align: center;">11</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">)</td><td style="text-align: center;">right parenthesis</td></tr></tbody></table></td> <td style="text-align: center;">13</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">=</td><td style="text-align: center;">numerical equality</td></tr></tbody></table></td> <td style="text-align: center;">15</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">x</td><td style="text-align: center;">numerical variable</td></tr></tbody></table></td> <td style="text-align: center;">17</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">y</td><td style="text-align: center;">numerical variable</td></tr></tbody></table></td> <td style="text-align: center;">19</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">z</td><td style="text-align: center;">numerical variable</td></tr></tbody></table></td> <td style="text-align: center;">23</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">...</td><td style="text-align: center;">numerical variable</td></tr></tbody></table></td> <td style="text-align: center;">29</td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">p</td><td style="text-align: center;">propositional variable</td></tr></tbody></table></td> <td style="text-align: center;">17<sup>2</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">q</td><td style="text-align: center;">propositional variable</td></tr></tbody></table></td> <td style="text-align: center;">19<sup>2</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">r</td><td style="text-align: center;">propositional variable</td></tr></tbody></table></td> <td style="text-align: center;">23<sup>2</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">...</td><td style="text-align: center;">propositional variable</td></tr></tbody></table></td> <td style="text-align: center;">29<sup>2</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">P</td><td style="text-align: center;">predicate variable</td></tr></tbody></table></td> <td style="text-align: center;">17<sup>3</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">Q</td><td style="text-align: center;">predicate variable</td></tr></tbody></table></td> <td style="text-align: center;">19<sup>3</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">R</td><td style="text-align: center;">predicate variable</td></tr></tbody></table></td> <td style="text-align: center;">23<sup>3</sup></td></tr><tr> <td><table border="1" style="width: 100%;"><tbody><tr><td style="text-align: center; width: 20%;">...</td><td style="text-align: center;">predicate variable</td></tr></tbody></table></td> <td style="text-align: center;">29<sup>3</sup></td></tr></tbody></table></center><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">The expression f(0) represents the successor of 0, that is, 1 = f(0) = 0 + 1; t</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">he expression f(f(0)) represents the successor of f(0), that is, 2 = f(1) = 1 + 1; t</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">he expression f(f(f(0))) represents the successor of f(f(0)), that is, 3 = f(2) = 2 + 1; etc. Therefore, any arbitrarily large integer can be represented in this system.</span><br /><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">These "numbers," of the form f( ... f(0)...), can be combined with the equality symbol to form propositions, which are either true or false. So, under the intended interpretation, the proposition 0 = 0 is true but the proposition 0 = f(0) is false.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">In turn, these propositions can be combined using the logical connectives OR and NOT. For example, under the intended interpretation, the compound proposition (0 = 0</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">)&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨ (0 = f(0)) is true, while&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">the compound proposition (&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(0 = 0</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">))&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨ (0 = f(0)) is false. Note how the left and right parentheses are used both around function arguments, like in f(0), and for grouping sub-sequences of symbols that go together.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">The last constant symbol we have yet to discuss is the "for all" (or universal) quantifier. Before we do so, we need to describe the first class of variables: the numerical variables x, y, z, x1, x2, etc., simply stand for unknown numerical expressions of the form f( ... f(0) ... ). Therefore, the formula&nbsp;</span><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(0 = x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">)&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;= f(x)))</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">is neither true nor false, since the value of x is unknown. We say that x is a&nbsp;<b>free variable</b> in this formula.&nbsp;</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">To eliminate this free variable, we could either replace every free occurrence of x in the formula by a "number" (this operation is called <b>substitution</b>) or &nbsp;we could add&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∀x in front of the formula to yield:&nbsp;</span><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∀x (&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(0 = x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">)&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;(</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;= f(x))</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">) )</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">In this new formula, x is not a free variable any longer because it appears within the "scope" of the universal quantifier, that is, between the two outermost parentheses. We say that x is <b>bound</b> by the quantifier. The intended meaning of this formula is the following: for every possible integer x, either x is equal to 0 or x is not equal to its successor, which is a true statement.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">The previous formula would have been false under the intended interpretation if the logical conjunction (OR) had been replaced with a logical conjunction (AND). Even though the logical connective AND is not explicitly part of G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del's formal system, it can easily be simulated using one of <a href="http://en.wikipedia.org/wiki/De_Morgan's_laws"><span class="Apple-style-span" style="color: blue;">de Morgan's laws</span></a>. In fact, the&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨ and</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ connectives are logically complete, which means that all of the other logical connectives (logical implication, equivalence, exclusive or, etc.) can all be simulated using only the two logical connectives in G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del's alphabet.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Another class of variables are the propositional variables p, q, r, p1, p2, etc., which stand for propositions such as 0 = 0,&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∀x </span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;= f(x)) ), and (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ </span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">p)&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∨</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;q.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Finally, the last class of variables are the predicates P, Q, R, P1, P2, etc., which stand for properties of integers. For examples, P(x) could stand for "x is a prime number" or "x is even." Note that we could have also considered predicates for relations between two or more integers such as Q(x,y) to mean "x is greater than y" or "x is divisible by y." Then, of course, we would need to include the comma in our alphabet of basic signs.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">I hope that this description of the alphabet as well as the examples of formulas just mentioned are sufficient to give a sense of what it means for a formula to be <b>well-formed</b>, that is, syntactically correct, in G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del's formal system. For example, the expression&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;0 is not a well-formed formula (or <b>wff</b>) since logical negation can only be applied to propositions, not to numbers. In contrast, both&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">0 = f(0) and&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">0 = 0 are wff's (but only the second one is true under the intended interpretation). This formal system is essentially first-order predicate calculus with equality, together with the basic vocabulary and axioms of integer arithmetic. We'll leave the axioms and inference rules for another time.</span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Now we can turn our attention to G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del's mapping between the set of wff's (the domain) and the set of non-negative integers (the co-domain).&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del's numbering scheme is the name given to the way he assigned a single integer to each wff (as well as to each sequence of wff's) of this formal system.&nbsp;</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">First, as shown in Table 1 above, G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del associated each basic sign or symbol in his alphabet with a unique integer. He mapped the constant signs to the first 8 odd integers (from 1 to 15). Then he&nbsp;</span>mapped the numerical variables to the prime numbers larger than 15. Note that there are an infinite number of such variables (including subscripted ones x1, x2, etc. if we needed to add those to the alphabet) and an infinite number of primes. Then he mapped the propositional variables to the squares of primes larger than 15. Finally, he mapped the predicate variables to the cubes of primes larger than 15.<br /><br />So, in the first part of the mapping just described, each alphabet symbol is mapped to a distinct integer.</div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span>Second,&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del mapped the set of wff's to a subset of the integers as follows. Each wff is made up of an (ordered) sequence of alphabet symbols. Any wff made up of n symbols, say the wff c<sub>1</sub></span>c<sub>2</sub>c<sub>3</sub>...c<sub>n</sub>, is mapped to the product of n integers, namely<br /><br /><div style="text-align: center;">2<sup>Φ(c<sub>1</sub>)</sup>&nbsp;×&nbsp;3<sup>Φ(c<sub>2</sub>)</sup>&nbsp;×&nbsp;5<sup>Φ(c<sub>3</sub>)</sup>&nbsp;×&nbsp;...&nbsp;×&nbsp;p<sub>n</sub><sup>Φ(c<sub>n</sub>)&nbsp;</sup></div><br />where&nbsp;p<sub>n</sub> is the n<sup>th</sup> prime number and Φ(c) is the Gödel number of the symbol c. Despite the complicated-looking notation, this mapping is quite simple. First, you must keep in mind the first few prime numbers (2,3,5,7,11,13,17,19,23, 29, 31, 37, 41, etc.), Second, you raise each prime to the power of the corresponding symbol in the wff. Finally, you multiply all of these powers of primes to get the Gödel number for the wff.<br /><br />For example, the wff 0 = 0 is made up of three symbols, whose Gödel numbers, from left to right, are 1, 15, and 1 (see Table 1). Therefore, the Gödel number of the wff 0 = 0 is<br /><br /><div style="text-align: center;">2<sup>1</sup>&nbsp;×&nbsp;3<sup>15</sup>&nbsp;×&nbsp;5<sup>1</sup></div><div style="text-align: center;"><span class="Apple-style-span" style="font-size: small;"><br /></span></div></div><div>A longer (but no more complicated) example is the mapping of the wff<br /><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">∀x </span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">¬ (</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;x</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;= f(x) ) )</span></div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">to its G</span>ödel number <span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><br /><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">2<sup>9</sup>&nbsp;×&nbsp;3<sup>17</sup>&nbsp;×&nbsp;5<sup>11</sup></span>&nbsp;× 7<sup>5</sup>&nbsp;× 11<sup>11</sup>&nbsp;× 13<sup>17</sup>&nbsp;× 17<sup>15</sup>&nbsp;× 19<sup>3</sup>&nbsp;× 23<sup>11</sup>&nbsp;× 29<sup>17</sup>&nbsp;× 31<sup>13</sup>&nbsp;×&nbsp;37<sup>13</sup>&nbsp;× 41<sup>13</sup></div><br />Third (and lastly),&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">G</span>ö<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">del mapped each sequence of wff's to a unique integer as follows. I will just illustrate the process with an example. Consider the following sequence of wff's:</span><br /><ul><li>x =&nbsp;0 whose Gödel number is&nbsp;2<sup>17</sup>&nbsp;×&nbsp;3<sup>15</sup>&nbsp;×&nbsp;5<sup>1</sup> (let's call this number α)</li><li>0 = 0&nbsp;whose Gödel number is&nbsp;2<sup>1</sup>&nbsp;×&nbsp;3<sup>15</sup>&nbsp;×&nbsp;5<sup>1&nbsp;</sup>(let's call this number β)</li><li><sup><span class="Apple-style-span" style="font-size: small;">p&nbsp;∨&nbsp;¬ p&nbsp;whose Gödel number is&nbsp;2</span><span class="Apple-style-span" style="font-size: small;"><sup>17<sup>2</sup></sup></span><span class="Apple-style-span" style="font-size: small;">&nbsp;×&nbsp;3</span><span class="Apple-style-span" style="font-size: small;"><sup>7</sup></span><span class="Apple-style-span" style="font-size: small;">&nbsp;×&nbsp;5</span><span class="Apple-style-span" style="font-size: small;"><sup>5</sup></span><span class="Apple-style-span" style="font-size: small;">&nbsp;× 7</span><span class="Apple-style-span" style="font-size: small;"><sup>17<sup>2</sup></sup></span>&nbsp;</sup>(let's call this number γ)</li></ul>The Gödel number for this sequence of 3 wff's is obtained by raising the first three primes to their Gödel numbers (respectively) and then multiplying them all together, yielding:<br /><br /><div style="text-align: center;">2<sup>α</sup>&nbsp;×&nbsp;3<sup>β</sup>&nbsp;×&nbsp;5<sup>γ</sup><br /><sup><br /></sup></div><div style="text-align: center;"><span class="Apple-style-span" style="font-size: small;">that is</span><br /><span class="Apple-style-span" style="font-size: small;"><br /></span></div><div style="text-align: center;">&nbsp;2<sup>(2<sup>17</sup>&nbsp;×&nbsp;3<sup>15</sup>&nbsp;×&nbsp;5<sup>1</sup>)</sup>&nbsp;×&nbsp;3<sup><span class="Apple-style-span" style="font-size: 14px;">(</span>2<sup>1</sup>&nbsp;×&nbsp;3<sup>15</sup>&nbsp;×&nbsp;5<sup>1</sup><span class="Apple-style-span" style="font-size: 14px;">)&nbsp;</span></sup>× 5<sup>(2<sup>17<sup>2</sup></sup>&nbsp;×&nbsp;3<sup>7</sup>&nbsp;×&nbsp;5<sup>5</sup>&nbsp;× 7<sup>17<sup>2</sup></sup>)</sup><br /><sup><br /></sup></div></div>Now, why did Gödel map not only the set of wff's but also the set of sequences of wff's to the set of integers? That's because each wff may (or may not) be a theorem of the formal system and each sequence of wff's may (or may not) be a proof of a theorem in the formal system. Gödel will use (and significantly extend) this mapping to complete his famous proof.<br /><br />But we are not there yet. This post has only explained the Gödel numbering scheme.&nbsp;This is all there is to it! Sure, the numbers get large. But that does not matter, since we are not going to compute their values. We just need to know that the mapping exists.<br /><br />However, there is one important property of this mapping that I have not mentioned yet.&nbsp;Recall that, in general, a mapping assigns to each element of the domain one or more elements of the co-domain. In this case, the mapping is really a <b>function</b>, since it assigns to each element of the domain exactly one element of the co-domain. &nbsp;But, more importantly, Gödel's function is injective.<br /><br />In an<b> injective </b>(or<b> one-to-one</b>)<b>&nbsp;</b>function, all of the elements of the domain are mapped to <i>distinct</i> elements of the co-domain.&nbsp;In the case of Gödel's function, the domain is the set of wff's&nbsp;"unioned" with the set of sequences of 2 or more wff's. The co-domain is the set of positive integers.<br /><div><br /></div>In other words, if you take any two distinct wff's or sequences of wff's, their Gödel numbers are guaranteed to be distinct integers. This is an immediate consequence of the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"><span class="Apple-style-span" style="color: blue;">fundamental theorem of arithmetic</span></a>, according to which, each integer greater than 1 can be decomposed into a unique product of powers of prime numbers.<br /><br />In fact, there is a relatively simple algorithm to convert a given integer to its corresponding wff or sequence of wff's, if it exists. But before we get to the algorithm, a few observations are in order:<br /><br /><ul><li>Not all integers are mapped to an element of the domain (in other words, Gödel numbering is not a<b> surjective</b> or <b>onto</b> function). For example, the number 20 =&nbsp;2<sup>2</sup>&nbsp;×&nbsp;5<sup>1</sup>&nbsp;cannot be a valid Gödel number since it is missing a power of 3. Only consecutive primes (starting at 2) can appear in a Gödel number. Another counterexample is&nbsp;3<sup>1</sup>&nbsp;×&nbsp;5<sup>2</sup>&nbsp;= <span class="Apple-style-span" style="font-size: small;">75, since the power of 2 is missing.</span></li><li><span class="Apple-style-span" style="font-size: small;">None of the numbers in Table 1 above are mapped to an element of the domain. By itself, no alphabet symbol is a wff, except for the propositional variables, such as p. But in this case, its Gödel number is&nbsp;</span>2<sup>(2<sup>17</sup>)</sup>, that is a one-symbol wff, as opposed to a single symbol whose Gödel number is 2<sup>17</sup>.</li><li>In order to prevent a wff w and the sequence made up of the single wff &nbsp;w (in essence, the same wff) from having two distinct Gödel numbers, we require sequences of wff's to contain at least two wff's.</li></ul>A simple recursive algorithm to convert any integer n into its corresponding&nbsp;wff or sequence of wff's (if it exists) is as follows:<br /><div><br /><ol><li>If n is less than 1 or is in Table 1 above, then no wff or sequence of wff's maps to it. <b>Terminate with failure</b>.</li><li>Compute the prime factorization of n.</li><li>If the primes in the factorization of n are not consecutive or do not start at 2, no wff or sequence of wff's maps to n. <b>Terminate with failure</b>.</li><li>If every exponent in the prime factorization of n is one of the numbers in Table 1 above, then convert the sequence of exponents in the prime factorization of n to the corresponding sequence of alphabet symbols.&nbsp;If the resulting sequence is a wff, <b>terminate and return the wff</b>.</li><li>If the prime factorization of n contains a single prime power, then <b>terminate with failure</b>.</li><li>Call this algorithm recursively on each exponent in the factorization of n. If all recursive calls return a wff, then n is mapped to this sequence of wff's. <b>Terminate and return this sequence of wff's</b>. Otherwise, <b>terminate with failure</b>.</li></ol><br />In conclusion, since wff's in Gödel's formal system are,&nbsp;at least under the intended interpretation,&nbsp;arithmetical propositions, they are part of mathematics. Therefore, Gödel numbering is a mapping between (formal) mathematical statements and integers.<br /><br />In future posts, we'll get into Gödel's proofs of the incompleteness theorems, which use a more sophisticated mapping involving meta-mathematical statements.</div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/yu-vohtxB00" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/06/mappings-and-godel-numbering.htmltag:blogger.com,1999:blog-6792710671733445593.post-49636491232069307222013-05-28T20:28:00.001-07:002013-06-01T08:56:18.041-07:00Absolute proof of consistency of FSNIn section IV of their great little book, Nagel and Newman discuss efforts by Gottlob Frege and then Bertrand Russell to reduce arithmetic to logic. This is clearly another attempt at a relative proof of consistency: if this reduction were successful, then arithmetic would be consistent provided logic is consistent.<br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Whether this latter statement is true or not, Whitehead and Russell's Principia Mathematica was a landmark achievement: it almost completed the first step in an absolute proof of consistency of arithmetic, since it led to the formalization of an axiomatic system for arithmetic.<br /><br />In section V, Nagel and Newman describe a formalization of propositional (or sentential) logic, that is, a subset of the logic system in&nbsp;Principia Mathematica (but not a large enough subset to represent arithmetic). The bulk of this section first describes the formalization process, which yields the standard syntax and inference rules of propositional logic (including <a href="https://en.wikipedia.org/wiki/Modus_ponens" target="_blank"><span class="Apple-style-span" style="color: blue;">modus ponens</span></a>) and then outlines an absolute proof of consistency of this formalized axiomatic system.<br /><br />This absolute proof of consistency is a&nbsp;<a href="http://en.wikipedia.org/wiki/Proof_by_contrapositive" target="_blank"><span class="Apple-style-span" style="color: blue;">proof by contrapositive</span></a>, which relies on the following true conditional statement:<br /><a name='more'></a><br /><blockquote class="tr_bq">"If the formal system is not consistent, then it can prove ALL of the syntactically legal formulas in the formal system."</blockquote>[<a href="http://en.wikipedia.org/wiki/Principle_of_explosion" target="_blank"><span class="Apple-style-span" style="color: blue;">Here</span></a> is a quick proof of this statement, which is classically called "ex falso quodlibet."]<br /><br />Now, the contrapositive of this statement, which must then also be true, is:<br /><blockquote class="tr_bq">"If the formal system cannot prove ALL of the&nbsp;syntactically legal formulas in the formal system, then it is&nbsp;consistent."</blockquote>[Aside: In logic, a conditional statement and its contrapositive are logically equivalent, which means that they are either both true or both false. In this case, they are both true.]<br /><br />Finally, if we can prove that "the formal system cannot prove ALL of the&nbsp;syntactically legal formulas", then we can use modus ponens and the contrapositive statement above to infer that the formal system is consistent.<br /><br />Therefore, Nagel and Newman wrap up section V by proving that all theorems of 'their' formal system are tautologies (they are always true under all possible substitutions of propositions for propositional variables) and that some syntactically legal formulas, such as p ∨ q, are not tautologies and are therefore not theorems of the formal system.<br /><br />In the rest of this post, instead of going over their proof in detail, I will provide a different and simpler, absolute consistency proof, namely the consistency of <a href="http://summerofgodel.blogspot.com/2013/05/absolute-proofs-of-consistency-and-meta.html" target="_blank"><span class="Apple-style-span" style="color: blue;">our simple formal system called <i>FSN</i></span></a>. &nbsp;Recall the axiom and inference rules of&nbsp;<i>FSN</i>:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Axiom 1 [A1]:</b></div></div></div></div></td><td><table border="1"><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>&nbsp;E0&nbsp;</i></div></div></div></div></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 1 [IR1]:</b></div></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>Eα&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;is any non-empty finite sequence of 0's and 1's)</span></span></div></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E0</i><i>α</i></div></div></div></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 2 [IR2]:</b></div></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E</i><i>α&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;is any non-empty finite sequence of 0's and 1's)</span></span></div></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E1</i><i>α</i></div></div></div></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 3 [IR3]:</b></div></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>Eβ0&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><i>β</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">&nbsp;is any finite sequence of 0's and 1's)</span></div></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>¬Eβ1</i></div></div></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Here is a really short sketch of the absolute proof of consistency of <i>FSN</i>&nbsp;that fills up the rest of this post: <i>FSN</i>&nbsp;cannot derive both a theorem and its negation because all of the theorems that start with a negation symbol end with the symbol '1' but all of the theorems that do not start with a negation symbol end with the symbol '0'.<br /><br />Nevertheless, it is interesting to try and write a rigorous proof. Notice how the proof sketch above is a clear example of meta-reasoning, that is, reasoning about (and outside of) the formal system itself.<br /><br />So, before we give an absolute proof of consistency of <i>FSN</i>, let's start with a simple example of meta-reasoning about <i>FSN</i>.<br /><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Let <i>T</i> be any theorem of <i>FSN</i>. Let's prove the following meta-theorem:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>[MT1]</b>&nbsp;&nbsp;Every theorem of&nbsp;<i>FSN</i>&nbsp;has the form '<i>E</i><i>γ</i>' or the form '<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>¬E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>γ</i></span>' where&nbsp;<i>γ</i>&nbsp;is a non-empty sequence of 0's and 1's.&nbsp;</div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i><br /></i></span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">To prove this meta-theorem about the&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">set of theorems of</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;<i>FSN</i>, we must reason outside of the system. More specifically, we can write a proof by structural induction. In other words, we can analyze the structure of the axiom and inference rules to prove that all theorems generated by any one of them has the property stated in MT1.</span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><hr /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Proof of MT1 by structural induction: Let<i>&nbsp;T</i>&nbsp;be any theorem of&nbsp;<i>FSN</i>.</span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">If&nbsp;<i>T</i>&nbsp;is produced by A1, then&nbsp;<i>T</i>&nbsp;is '<i>E0</i>' which has the form '<i>E</i><i>γ</i>'; therefore&nbsp;<i>T</i>&nbsp;has one of the forms required by the theorem.</span></li><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">If&nbsp;<i>T</i>&nbsp;is produced by IR1, then&nbsp;<i>T</i>&nbsp;has the form&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">'<i>E0α</i>'&nbsp;</span>which&nbsp;has the form '<i>Eγ</i>', that is, one of the forms<i>&nbsp;</i>required by the theorem.</li><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">If&nbsp;<i>T</i>&nbsp;is produced by IR2, then&nbsp;<i>T</i>&nbsp;has the form&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">'<i>E1α</i>'&nbsp;</span>which&nbsp;has the form '<i>Eγ</i>', that is, one of the forms<i>&nbsp;</i>required by the theorem.</li><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">If&nbsp;<i>T</i>&nbsp;is produced by IR3, then&nbsp;<i>T</i>&nbsp;has the form&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">'<i>¬Eβ1</i>'&nbsp;</span></span></span>which&nbsp;has the form '<i><span class="Apple-style-span" style="font-style: normal;"><i>¬</i></span>Eγ</i>', that is, one of the forms<i>&nbsp;</i>required by the theorem.</li></ul><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Since<i>&nbsp;T</i>&nbsp;has the desired property in all cases, MT1 holds.<br />□</div><hr /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br />Notice that we did not need an inductive hypothesis in this proof because the conclusion of each inference rule clearly satisfies the needed property regardless of whether its premise does.</div></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Now, let's define three properties <i>P, Q</i> and <i>R</i>&nbsp;of <i>T</i>. We say that:</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><ul><li><i>P(T)</i> is true if<i> T</i> starts with the symbol 'E' and ends with the symbol '0'.</li><li><i>Q(T)</i>&nbsp;is true if<i>&nbsp;T</i>&nbsp;starts with the symbol '<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>¬</i></span>' and ends with the symbol '1'.</li><li><i>R(T)</i> is true if and only if exactly one of <i>P(T)</i> and <i>Q(T)</i> is true.</li></ul><div>We can easily prove the following meta-theorem about <i>FSN</i>:</div><br /><b>[MT2]</b>&nbsp;&nbsp;For every theorem <i>T</i> of&nbsp;<i>FSN</i>, <i>R(T)</i> is true.<br /><br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><hr /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Proof of MT2 by structural induction: Let<i>&nbsp;T</i>&nbsp;be any theorem of&nbsp;<i>FSN</i>.</span></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Basis step: <i>T</i>&nbsp;is produced by A1</span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>T</i></span>&nbsp;is '<i>E0</i>' which starts with the symbol 'E' and ends with the symbol '0'; therefore&nbsp;<i>R(T)</i>&nbsp;is true in this case.</li></ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">First inductive step: <i>T</i>&nbsp;is produced by IR1</span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Inductive hypothesis: Assume that <i>R(E</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i>&nbsp;is true</span></span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>Q(E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i>&nbsp;must be false since '<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i>' does not start with&nbsp;</span></span></span>'<i>¬</i>'. Thus&nbsp;<i>P(E</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i></span>&nbsp;must be true and&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;must end with the symbol '0'. Therefore, &nbsp;<i>T</i>&nbsp; has the form&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">'E0<i>α</i>' which starts with the symbol 'E' and ends with the symbol '0'; therefore <i>R(T)</i>&nbsp;is true in this case.</span></li></ul></ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Second inductive step:&nbsp;<i>T</i>&nbsp;is produced by IR2</span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Inductive hypothesis: Assume that&nbsp;<i>R(E</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i>&nbsp;is true</span></span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>Q(E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i>&nbsp;must be false since '<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i>' does not start with&nbsp;</span></span></span>'<i>¬</i>'. Thus&nbsp;<i>P(E</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α)</i></span>&nbsp;must be true and&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;must end with the symbol '0'. Therefore, &nbsp;<i>T</i>&nbsp; has the form&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">'E1<i>α</i>' which starts with the symbol 'E' and ends with the symbol '0'; therefore&nbsp;<i>R(T)</i>&nbsp;is true in this case.</span></li></ul></ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Third inductive step:&nbsp;<i>T</i>&nbsp;is produced by IR3</span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Inductive hypothesis: Not needed here</span></li><ul><li><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>T &nbsp;</i>must have the form<i>&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>¬Eβ1</i></span></i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">. Thus <i>Q(T) </i>is true and, therefore,&nbsp;<i>R(T)</i>&nbsp;is also true in this case.</span></li></ul></ul></ul><div>Since<i>&nbsp;R(T)</i>&nbsp;is true for all theorems <i>T</i>&nbsp;of&nbsp;<i>FSN</i>,&nbsp;MT2 holds.<br />&nbsp;□</div><hr /><div><br /></div><div>Finally, it is only one short step from MT2 to ...</div><div><br /></div><div><b>[MT3]</b> <i>FSN</i> is consistent.</div><div><br /></div><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><hr /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Proof of MT3 by contradiction:</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">Assume that <i>FSN</i> is not consistent.</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">By definition of (in)consistency, there must exist two theorems of <i>FSN</i> of the form <i>T</i> and&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>¬</i></span>T</i>. Since&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>¬</i></span>T</i> starts with a negation symbol, this theorem must, by MT1, have the form&nbsp;&nbsp;'<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>¬E</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>γ</i></span>' where&nbsp;<i>γ</i>&nbsp;is a non-empty sequence of 0's and 1's. According to MT2,&nbsp;<i>γ</i>&nbsp;must end with the symbol '1' (to satisfy property <i>Q</i>). Therefore,&nbsp;<i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>¬</i></span>T&nbsp;</i>must have the form&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>¬E</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>β</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>1. </i>But then</span>&nbsp;<i>T</i> must have the form&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>β</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>1</i></span>, which contradicts MT2 (since neither <i>P(</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>β</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>1</i></span><i>)</i> nor<i> Q(</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E</i></span><i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-style: normal;"><i>β</i></span></i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>1</i></span><i>)</i> is true).&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">In conclusion, <i>FSN </i>is consistent.</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">&nbsp;□</div></div></div><hr /><div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/LaR1ZmGz-WA" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/05/absolute-proof-of-consistency-of-fsn.htmltag:blogger.com,1999:blog-6792710671733445593.post-54365279247265039852013-05-26T20:34:00.002-07:002013-06-01T10:23:18.405-07:00Absolute proofs of consistency and meta-mathematicsIn earlier posts, we explained why consistency is an important property of axiomatic systems and discussed relative proofs of consistency, in which the proof of consistency of a system is based on the assumption that another axiomatic system is consistent. In this post, we introduce absolute proofs of consistency that do not make any assumptions about any other axiomatic system. Apparently, <a href="http://en.wikipedia.org/wiki/David_Hilbert" target="_blank"><span class="Apple-style-span" style="color: blue;">David Hilbert</span></a> was the first to study and propose such proofs, according to Nagel and Newman's book (Section III, page 26).<br /><br />Recall that an axiomatic system is consistent if it cannot derive both a theorem and its negation. What do we mean by the negation of a theorem? Let's take, as a simple example, the theorem: "6 is divisible by 3." Its negation is simply the following statement: "6 is not divisible by 3" or equivalently "It is not the case that 6 is divisible by 3." This second formulation of the negation, although less elegant in English, is preferable because the negation is added to the front of the original theorem. In a formal system, negation is handled by simply adding a symbol for the phrase "it is not the case that." Several symbols have been used for negation, such as ~ and ¬ . We'll use the latter here. So, if<i> T</i> is any theorem in some formal system, then the formula&nbsp;&nbsp;¬<i>T </i>is the negation of <i>T</i>.<br /><br />Remember that our formal system <a href="http://summerofgodel.blogspot.com/2013/05/formal-systems-axioms-inference-rules.html" target="_blank"><span class="Apple-style-span" style="color: blue;"><i>FS</i></span></a> did not have a symbol for negation. So we will extend <i>FS</i> into a new formal system called&nbsp;<i>FSN</i>&nbsp;(for&nbsp;<i>FS</i>&nbsp;with<i>&nbsp;N</i>egation), whose alphabet is { <i>E</i>,<i> 0</i>, <i>1</i>,&nbsp;¬&nbsp;}. <i>FSN</i> has exactly one axiom, namely the same as A1 in<i> FS</i>.&nbsp;<i>FSN</i>&nbsp;also has the same inference rules as <i>FS</i>, namely IR1 and IR2. But it has one additional inference rule that uses negation. Here is the full description of <i>FSN</i>:<br /><br /><a name='more'></a><br /><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Axiom 1 [A1]:</b></div></div></div></td><td><table border="1"><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>&nbsp;E0&nbsp;</i></div></div></div></td></tr></tbody></table></td></tr></tbody></table><br /><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 1 [IR1]:</b></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>Eα&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;is any non-empty finite sequence of 0's and 1's)</span></span></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E0</i><i>α</i></div></div></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><br /><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 2 [IR2]:</b></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E</i><i>α&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">&nbsp;is any non-empty finite sequence of 0's and 1's)</span></span></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E1</i><i>α</i></div></div></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><br /><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 3 [IR3]:</b></div></div></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>Eβ0&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">(where&nbsp;</span></span><i>β</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">&nbsp;is any finite sequence of 0's and 1's)</span></div></div></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div></div></div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>¬Eβ1</i></div></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><br /><br />Notice that the new rule of inference takes any theorem (the premise) starting with <i>E</i> and ending in<i> 0</i> and produces a new theorem that results from adding a negation symbol in front of the <i>E</i> and replacing the final <i>0</i> with a <i>1</i>.<br /><br />Can <i>FSN</i>&nbsp;produce all of the theorems that<i> FS</i> produces? Does&nbsp;<i>FSN</i>&nbsp;produce new theorems? If so, which form do these new theorems have? And, most importantly, ...<br /><br />... is<i> FSN</i> consistent?<br /><br />And if so, can we come up with an absolute proof of consistency?<br /><br />The first step in an absolute proof, as envisioned by Hilbert, is to completely formalize the axiomatic system whose consistency we want to prove. Since <i>FSN</i> is already formalized, we can skip this step here.<br /><br />However, classical axiomatic systems, such as Euclidean geometry, were not formalized. As shown above, a formal system is comprised of axioms and inference rules, each of which is built out of strings of meaningless symbols according to purely formal or syntactical rules. The main advantage of formalized axiomatic systems is that&nbsp;all logical steps are explicit and no hidden assumptions can creep in. In contrast, Euclidean geometry stated its axioms and theorems using informal and meaningful language (Greek). In addition, its inference rules were not formalized, allowing for unstated assumptions and implicit inferences. Since these are not allowed in a rigorous proof, we must get rid of them by formalizing the axiomatic system.<br /><br />The second step in an absolute proof of consistency is to prove that the formalized axiomatic system can never produce&nbsp;(i.e., prove)&nbsp;a theorem and its negation. Since the way theorems are produced is completely mechanical, based on explicit and unambiguous symbol manipulations, it may be possible to build a rigorous proof of consistency by studying the formal system from the outside.<br /><br />At this point, it is important to <b>distinguish reasoning that takes place within the formal system from reasoning that takes place outside the system</b>.<br /><br />Reasoning within the system is limited to asserting axioms and applying inference rules. This reasoning, as discussed before, is purely syntactic and meaningless. Here are a few "sentences" that can be generated within the system:<br /><ol><li>E0</li><li>E00</li><li>E10</li><li>¬ E1</li><li>¬ E101</li></ol><div>Now, what kind of reasoning are we going to need to prove that our formal system&nbsp;<i>FSN</i> is consistent (assuming that it is)? Clearly, the theorems of <i>FSN</i> cannot prove the consistency of <i>FSN</i>. The theorems of <i>FSN</i> are meaningless strings of symbols. They are not about anything. They cannot say anything about <i>FSN</i>. Therefore, they cannot say that <i>FSN</i> is consistent.</div><div><br /></div><div>To prove the consistency of <i>FSN</i>, we need to reason about the whole set of theorems of <i>FSN</i>. If the proofs obtained within <i>FSN</i>&nbsp;constitute a kind of (formal) reasoning, then reasoning <i>about</i> this reasoning is called <b><i>meta</i>-reasoning</b>. Proving that <i>FSN</i> cannot prove both a theorem and its negation requires meta-reasoning. Here are some examples of statements about <i>FSN</i> that might result from meta-reasoning:</div><div><ol><li>The theorem 'E110' is derivable from the axiom 'E0' by two applications of IR2.</li><li>The symbol 'E' appears exactly once in each theorem of <i>FSN</i>.</li><li>'¬E100' is not a theorem of <i>FSN</i>.</li><li><i>FSN</i> is consistent.</li><li><i>FSN</i> is not consistent.</li></ol></div><div>Hopefully, by now, you have figured out which one of statements 4 and 5 above is true.<br /><br />Now, we can step back from our made-up formal system and consider the kind of formal systems that Hilbert was interested in. These formal systems, of course, focused on mathematics and included Whitehead and Russell's <a href="http://en.wikipedia.org/wiki/Principia_Mathematica" target="_blank"><span class="Apple-style-span" style="color: blue;">Principia Mathematica</span></a>&nbsp;as well as&nbsp;<a href="https://en.wikipedia.org/wiki/Peano_axioms" target="_blank"><span class="Apple-style-span" style="color: blue;">Giuseppe Peano's formal system for arithmetic</span></a>.&nbsp;The reasoning within these formal systems model mathematical reasoning. Therefore, statements <i>about</i> these formal systems are <i>about</i> mathematics. They are part of <b><i>meta</i>-mathematics</b>. So, the question whether one of these systems is consistent is part of meta-mathematics.<br /><br />Let's wrap-up with one more important feature of these formal systems: they contain a finite number of axioms (or axiom schemata) and a finite number of inference rules (or reasoning patterns). Why does this matter? <a href="http://summerofgodel.blogspot.com/2013/05/relative-proofs-of-consistency.html" target="_blank"><span class="Apple-style-span" style="color: blue;">Remember</span></a> how the model-based approach to proving consistency suffered from the impossibility of inspecting all of the elements of an infinite model (e.g., a model for arithmetic would have to include an infinite set of elements to be mapped to the set of natural integers).<br /><br />With the absolute approach proposed by Hilbert, there is no need for a model. Instead, in this framework, the proof of consistency is based solely on the study of a formal system, which is fully defined by a finite number of patterns of reasoning. This is the crux of the argument. Reasoning about an infinite domain of knowledge (such as arithmetic) is reduced to the study of a finite formal system. If possible, this approach would satisfy Hilbert's quest for so-called <a href="http://en.wikipedia.org/wiki/Finitary" target="_blank"><span class="Apple-style-span" style="color: blue;">finitary or finitistic methods</span></a>.</div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/B2fhEZtxBLs" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/05/absolute-proofs-of-consistency-and-meta.htmltag:blogger.com,1999:blog-6792710671733445593.post-91837568166257401772013-05-22T18:17:00.001-07:002013-06-01T10:24:17.056-07:00Relative proofs of consistencyIn the <span id="goog_2069030133"></span><a href="http://summerofgodel.blogspot.com/2013/05/consistency-of-axiomatic-systems.html" target="_blank"><span class="Apple-style-span" style="color: blue;">last&nbsp;post</span></a><span id="goog_2069030134" style="color: blue;"></span>, we defined and explained the importance of the property of consistency for an axiomatic system. The second half of Section II in Nagel and Newman's book describes ways of proving that an axiomatic system is consistent.<br /><br />But first, note that the question of consistency of Euclidean geometry did not arise. Its axioms were supposed to describe the real world; and something that actually exists cannot be self-contradictory. In other words, existence (or truth) implies internal consistency.<br /><br />The need for consistency proofs arose much later, with non-Euclidean geometries, which do not obviously model space as we experience it. Non-existence does not imply inconsistency. But any interesting abstract construct had better be internally consistent.<br /><br />Second, checking the internal consistency of all of the theorems produced so far is typically not a valid proof of consistency, because (interesting) axiomatic systems generate an infinite number of theorems. The proof of consistency must guarantee that not a single theorem, including some that we have not yet produced and that we might never produce, contradicts any other theorem in the system.<br /><br />One possible way to prove the consistency of an axiomatic system is model-based, where a model is a kind of interpretation.<br /><br /><a name='more'></a><br />An <b>interpretation</b>&nbsp;is a way to give meaning to an axiomatic system. More precisely, it is a mapping between the symbols of the system and entities that exist in a domain outside the system.<br /><br />For example, one interpretation of our formal system <a href="http://summerofgodel.blogspot.com/2013/05/formal-systems-axioms-inference-rules.html"><i><span class="Apple-style-span" style="color: blue;">FS</span></i></a> is the following:<br /><ul><li>the symbol '0' maps to the binary digit zero (the 0 bit)</li><li>the symbol '1' maps to the binary digit one (the 1 bit)</li><li>the symbol 'E' maps to the property of being an even integer</li><li>In this example, we will take the domain of the interpretation to be the set of non-negative integers.&nbsp;</li><li>More precisely, in <i>FS</i>, the string <i>Ex </i>(where <i>x</i> is a finite, non-empty sequence of 0's and 1's), is interpreted as the statement: "The non-negative integer whose binary representation is <i>x</i> is even." &nbsp;If you need a refresher on the binary representation of integers, click <a href="http://www.swarthmore.edu/NatSci/echeeve1/Ref/BinaryMath/NumSys.html"><span class="Apple-style-span" style="color: blue;">here</span></a>.</li></ul><br />A&nbsp;<b>model</b>, then,&nbsp;is an interpretation that makes all of the axioms in the system true.&nbsp;Of course, it is assumed that all of the inference rules are truth-preserving with respect to the model; that is, if the axioms are true, then so are all of the theorems derived from them within the system.<br /><br />Now two interesting questions arise:<br /><ol><li>Is the interpretation given above a model of <i>FS&nbsp;</i>?</li><li>How would you justify your answer to the previous question?</li></ol>If you answered 'YES' to the first question, then, according to the model-based approach to consistency, <i>FS</i> is consistent.<br /><br />Why?<br /><br />Here is a possible argument:<br /><br /><b>[Premise 1] </b>Under the given interpretation,&nbsp;<i>FS</i>&nbsp;perfectly mirrors a domain that exists outside of the axiomatic system.<br /><br /><b>[Premise 2]</b>&nbsp;This domain exists (or is true), and is thus consistent; recall that existence (or truth) implies consistency.<br /><br /><b>[Conclusion]</b>&nbsp;<i>FS</i> is consistent.<br /><br />Unfortunately,&nbsp;the model-based approach exhibits&nbsp;at least two weaknesses that are illustrated in the two premises above, respectively:<br /><br /><ul><li>Our model has an infinite domain (the set of non-negative integers). So how can we prove that <i>FS</i>&nbsp;really maps onto this domain, as stated in premise 1? In other words, how can we answer question 2 above? We certainly cannot do it by inspection, since we cannot possibly inspect an infinite number of elements. Given the extreme simplicity of <i>FS</i>, it is not surprising that axiomatic systems for full branches of mathematics (such as set theory, algebra, Euclidean and non-Euclidean geometries) typically do not have finite models. Unfortunately, intuitions about infinity are often misleading and do not protect from&nbsp;<a href="http://en.wikipedia.org/wiki/Russell's_paradox"><span class="Apple-style-span" style="color: blue;">well-known contradictions or antinomies</span></a> (i.e., inconsistencies). This is why David Hilbert developed a research program aimed at developing so-called<span class="Apple-style-span" style="color: blue;"> <a href="http://en.wikipedia.org/wiki/Hilbert's_program#Hilbert.27s_program_after_G.C3.B6del"><span class="Apple-style-span" style="color: blue;"><i><b>finitary</b></i> proof techniques</span></a></span>.</li></ul><ul><li>In our example, the model is (a subset of) arithmetic. But in what sense does arithmetic exist? Why is it true, and thus consistent, as stated in premise 2? Shouldn't these questions be settled first? What the model-based approach really does is reduce the consistency of <i>FS</i> to the consistency of (a subset of) arithmetic. Therefore, this proof of consistency of <i>FS</i> is a <b>relative proof</b>. It states that if arithmetic is consistent, then <i>FS</i> is also consistent. Contrast this with the (absolute) statement: <i>FS</i> is consistent. This is what we really wanted to prove.</li></ul><br />Nagel and Newman discuss other relative proofs of consistency, namely the proof given by Riemann of his non-Euclidean geometry relative to Euclidean geometry, as well as the proof given by Hilbert of Euclidean geometry relative to algebra.<br /><br />Clearly, the model-based approach is not the final word on proofs of consistency. Next time, we'll address these issues when we discuss the following section in the book.<img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/8qdTrV8M5Sk" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/05/relative-proofs-of-consistency.htmltag:blogger.com,1999:blog-6792710671733445593.post-81806054595569370572013-05-20T21:36:00.000-07:002013-06-01T10:25:24.992-07:00Consistency of axiomatic systemsThe "problem of consistency" is the topic of Section II of Nagel and Newman's book. &nbsp;This section defines "consistency" and explains when and why it became an important property of axiomatic systems.<br /><br />The oldest and most famous axiomatic system is that of Euclid, in which he systematized all of the knowledge of geometry (and more) available to him over two thousand years ago. Based on five axioms, Euclid was able to rigorously prove a very large number of known and new theorems (called "propositions" in his <i><a href="http://aleph0.clarku.edu/~djoyce/java/elements/elements.html"><span class="Apple-style-span" style="color: blue;">Elements</span></a></i>). His axioms were supposed to be intuitively true. The first four axioms dealt with line segments, lines, circles and angles (see this <a href="http://en.wikipedia.org/wiki/Euclidean_geometry#Axioms"><span class="Apple-style-span" style="color: blue;">Wikipedia entry</span></a>) and have been viewed as self-evident. In contrast, the fifth axiom, which was equivalent to the following statement: "Through a point outside a given line, only one parallel to the line can be drawn" (page 9), was not intuitively true (apparently because the two lines involved extend to infinity in two directions, similarly to asymptotes).<br /><br />Since this proposed axiom was not obviously true, many mathematicians tried to prove that it logically follows from the first four axioms. Only in the nineteenth century was it demonstrated that it is NOT possible to prove the parallel axiom from the first four axioms.<br /><br />This <b>proof that it is impossible to prove</b>&nbsp;a given statement is a great precursor of Gödel's incompleteness theorems.<br /><br /><a name='more'></a><br />Since the parallel "axiom" is logically independent of the first four axioms, it could be assumed to be either true or false. Euclidean geometry, which assumes that it is true, appears to be a good match for (or model of) our every day experience of 3-dimensional space. However,&nbsp;&nbsp;several mathematicians also developed alternate geometries that take the parallel "axiom" to be false (e.g., there is no parallel or there is more than one parallel line). These non-Euclidean geometries are not intuitively "true" since they do not seem to model our&nbsp;every day experience of 3-dimensional space. Yet, they appear to make complete sense, mathematically.<br /><br />But how can a mathematical system "make complete sense" if it is not obviously true? Well, what matters is that it <i>could</i> be true. In other words, the system must not be self-contradictory; it may not produce theorems that are mutually incompatible, such as one statement and its negation. If an axiomatic system meets this requirement, it is said to be <b>consistent</b>.<br /><br />The development of non-Euclidean geometries led to a major shift in the perception, if not the role, of axioms. Axioms were and remain the essential, logical foundation of all knowledge (i.e., theorems) in an axiomatic system. However, axioms did not have to be obviously true of the natural world. They just had to be mutually consistent.<br /><br />First, this shift gave much greater freedom to mathematicians, who, for example, had never before considered the possibility of non-Euclidean geometries.<br /><br />Second, this shift had a much greater impact on the discipline of mathematics as a whole. Mathematics became more abstract. Internally consistent systems became as interesting and important as actual models of the real world.<br /><br />Hence&nbsp;Bertrand Russell's famous quote:<br /><blockquote class="tr_bq"><span class="Apple-style-span" style="font-family: sans-serif; font-size: 14px; line-height: 21px;">"</span><span class="Apple-style-span" style="line-height: 21px;"><span class="Apple-style-span" style="font-family: inherit;">Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."</span></span></blockquote>But if internal consistency is enough, and the truth of the axioms (and therefore of the theorems as well) is only secondary, what is left? The short answer: the rigorous logical reasoning that leads from axioms to theorems.<br /><br />For even if the literal truth of the whole edifice is not required any longer, it must still stand firmly and securely on its foundations. In other words, each and every theorem of an axiomatic system <i>must necessarily</i> follow from the axioms. <br /><br />To establish the validity of logical inferences requires eliminating all sources of vagueness and making all steps in the reasoning fully explicit. But natural languages, presuppositions, intuition, etc., get in the way of precision and rigor. &nbsp; Therefore, mathematics did not just become more abstract, it became more formal.<br /><br />Pure mathematics became a symbols game. &nbsp;Contrast Euclidean geometry (with, for example, its axioms stated in natural language, in this case ancient Greek) to our formal system <i><a href="http://summerofgodel.blogspot.com/2013/05/formal-systems-axioms-inference-rules.html"><span class="Apple-style-span" style="color: blue;">FS</span></a></i>, where each axiom, each premise and each conclusion of each inference rule is a meaningless sequence of symbols. Reasoning in a formal system depends only on the "form" or structure of the symbolic expression. &nbsp;And so&nbsp;"axiomatic systems" became "<b>formal</b> axiomatic systems."<br /><br />The price to pay to guarantee that theorems logically and necessarily follow from axioms and other premises is to reduce logical inference to mechanical symbol manipulation, and most importantly, to&nbsp;give up meaning.<br /><br />What do the symbols <i>E</i>&nbsp;, <i>0</i>, and <i>1</i>&nbsp;in <i>FS</i> actually mean? It is not relevant. In fact, even if they do have a meaning (not within the system, of course, but for the human being who is studying the system from the outside), it is best to completely forget about it.<br /><br />The only guaranteed way to make logically valid inferences, that is, 100% error-free symbol manipulations, is to empty one's mind of any interpretation of the symbols that could lead to implicit presuppositions, bias, or misguided intuition, in other words, to act like a mechanical automaton or a digital electronic computer.<br /><br />Now, remember, we gave up the need for truth and meaning. But for the whole edifice to stand up to logical scrutiny, the formal system must still be consistent.<br /><br />In the next few posts, we will look at several ways of demonstrating the consistency of an axiomatic system.<br /><br /><br /><br /><br /><br /><br /><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/nYu_nQ7Y0LI" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/05/consistency-of-axiomatic-systems.htmltag:blogger.com,1999:blog-6792710671733445593.post-46083547870830631252013-05-17T18:00:00.000-07:002013-06-01T10:25:40.980-07:00Formal systems, axioms, inference rules, formal proofsI'll start this post by describing a simple formal system that I made up.<br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">A&nbsp;<b>formal system</b>&nbsp;is comprised of axioms and inference rules. Each axiom and inference rule is defined syntactically, that is, by ordered sequences of symbols that follow strict syntactic rules but do not (necessarily) have any meaning. The set of all symbols allowed in a formal system are explicitly listed in its&nbsp;<b>alphabet</b>, simply, a finite, non-empty set of symbols.</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">My formal system, let's call it&nbsp;<i>FS</i>, &nbsp;uses the alphabet {<i>E,0,1</i>} and has only one axiom and two inference rules. Here is the axiom (in the box below):</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /><a name='more'></a></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Axiom 1 [A1]:</b></div></td><td><table border="1"><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>&nbsp;E0&nbsp;</i></div></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">And here are the two inference rules in&nbsp;<i>FS </i>(in the boxes below):</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 1 [IR1]:</b></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>Eα</i></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">(where&nbsp;<i>α</i>&nbsp;is any non-empty finite sequence of 0's and 1's)</div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E0</i><i>α</i></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><b>Inference rule 2 [IR2]:</b></div></td><td><table border="1"><tbody><tr><td><table><tbody><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E</i><i>α</i></div></td><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">(where&nbsp;<i>α</i>&nbsp;is any non-empty finite sequence of 0's and 1's)</div></td></tr><tr><td><hr /></td><td></td></tr><tr><td><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><i>E1</i><i>α</i></div></td><td></td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><br /></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">It is important to realize that, in the two inference rules above, the symbol&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;is NOT an acceptable sequence of symbols in<i>&nbsp;FS.&nbsp;</i>It cannot be, since&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;does not belong to the alphabet of<i>&nbsp;FS.</i>&nbsp;Instead,<i>&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;is a placeholder that stands for any&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">non-empty finite sequence of 0's and 1's. So, for example,&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;could stand for 0. Or it could stand for 1. Or it could stand for 1101. In fact, in each application of the inference rules above,&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;will stand for a single finite sequence of 0's and 1's chosen from the infinite set of such sequences.&nbsp;</div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">But what do we mean by an "application" of an inference rule? To answer this question, let's first discuss what formal systems are used for. The goal of a formal system is to formalize or mechanize the proofs of a well-defined set of theorems. &nbsp;Each sequence (or string) of symbols produced by the formal system is a&nbsp;<b>theorem</b>.&nbsp;</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">There are two ways for a formal system to "produce" or "prove" a theorem: using an axiom or using an inference rule.&nbsp;</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">First, an axiom is,&nbsp;<i>by definition</i>, a theorem. So, going back to our&nbsp;<i>FS</i>, its first theorem is the string&nbsp;<i>E0</i>. Why? Because&nbsp;<i>E0</i>&nbsp;is an axiom of&nbsp;<i>FS</i>! That is the proof of&nbsp;<i>E0</i>&nbsp;in&nbsp;<i>FS</i>. There is no other justification needed. Each axiom of a formal system is a theorem of that formal system.</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Second, and more interestingly, theorems can be produced (or proved) using inference rules and already proven theorems. As you can tell from the two examples above, an inference rule has two parts, separated by a horizontal line. Above the horizontal line comes the <b>premise</b> of the inference rule (there could be more than one premise). Below the horizontal line is the <b>conclusion</b> of the inference rule (there is exactly one conclusion). Each inference rule can be interpreted as: "If the premise is a theorem of <i>FS</i>, then the conclusion is also a theorem." So each application of an inference rule is a way of proving a new theorem given previously proven theorems.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Let's illustrate this point with several applications of the rules [IR1] and [IR2] in <i>FS</i>.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Assume that we start the process with an empty set of theorems, denoted by {&nbsp;}.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">At this point, neither [IR1] nor [IR2] can be applied because we have not yet proved any theorems, let alone any theorems that match the form of their premise. However, we do have an axiom to kick start the theorem proving process. Indeed,&nbsp;</span>axiom [A1] proves the theorem <i>E0</i>.<br /><br />After this first proof, our set of theorems is as follows: {&nbsp;<i>E0&nbsp;</i>}<br /><br />Now that our set of theorems is not empty any more, we can check if any of its elements matches the premises of [IR1] or [IR2]. It turns out that <i>E0</i> does match the premise of [IR1], since 1) both&nbsp;<i>E0&nbsp;</i>and&nbsp;<i>E</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i></span>&nbsp;start with the symbol <i>E</i> and 2) 0, being a non-empty <span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">finite sequence of 0's and 1's,</span>&nbsp;can replace&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i>. Since the premise of [IR1] is true (that is, it is the theorem of <i>FS&nbsp;</i>that we prove in the previous step), we can apply the inference rule [IR1] and, doing so, we prove its conclusion, namely the theorem <i>E00. </i>How did we obtain this new theorem? We started with&nbsp;<i>E0</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i>&nbsp;and replaced<i>&nbsp;</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i>&nbsp;with 0, which was the match in the premise:&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α</i> must be replaced uniformly throughout all parts of the inference rule being applied.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">After this second proof, our set of theorems is as follows: {&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E0</i> , <i>E00&nbsp;</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">}.</span></span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><br /></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">What choices do we have for the next proof? We could:</span></span><br /><ol><li>Use axiom [A1]</li><li>Apply [IR1] to <i>E0</i></li><li>Apply [IR2] to&nbsp;<i>E0</i> (with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 0)</li><li><i><span class="Apple-style-span" style="font-style: normal;">Apply [IR1] to</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>E00&nbsp;</i></span></i>(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 00)</li><li><i><span class="Apple-style-span" style="font-style: normal;"><i><span class="Apple-style-span" style="font-style: normal;">Apply [IR2] to</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>E00&nbsp;</i></span></i></span></i>(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 00)</li></ol><div>Since we already used the first two options, there is no point in choosing them again: each would yield a theorem that we have already proved. But we can pick any one of the three other options. Let's choose option 5. Its premise matches <i>E00</i> with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 00; therefore its conclusion yields the theorem&nbsp;<i>E100</i>.</div><br />After this third proof, our set of theorems is as follows: {&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E0</i>&nbsp;,&nbsp;<i>E00</i></span>&nbsp;,&nbsp;<i>E100</i><i>&nbsp;</i><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">}.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"></span><br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">What UNUSED options do we have for the next proof? We could:</span></span></div><ol><li>Apply [IR2] to&nbsp;<i>E0</i>&nbsp;(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 0)</li><li><i><span class="Apple-style-span" style="font-style: normal;">Apply [IR1] to</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>E00&nbsp;</i></span></i>(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 00)</li><li><i><span class="Apple-style-span" style="font-style: normal;"><i><span class="Apple-style-span" style="font-style: normal;">Apply [IR1] to</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>E100&nbsp;</i></span></i></span></i>(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 100)</li><li><i><span class="Apple-style-span" style="font-style: normal;"><i><span class="Apple-style-span" style="font-style: normal;">Apply [IR2] to</span><span class="Apple-style-span" style="font-style: normal;">&nbsp;</span><span class="Apple-style-span" style="font-style: normal;"><i>E100&nbsp;</i></span></i></span></i>(with&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>α&nbsp;</i></span>replaced by 100)</li></ol><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">Why don't you pick one of these options and determine which new theorem it proves? (the answers are given below) As you can tell, the number of theorems in our set increases fast. To every new theorem we prove, we can apply the two inference rules of <i>FS</i> to produce two new theorems.</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">So, how many theorems does&nbsp;<i>FS</i>&nbsp;generate in total?</span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">That's right! This simple formal system can prove an infinite number of theorems, even though&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;">each one of these theorems is comprised of a finite number of symbols from the alphabet of <i>FS</i>. By the way, here are the theorems produced by the four options above, in order: <i>E10</i>,&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E000</i>,&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E0100</i>, and&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E1100</i>.</span><br /><br />So, after these four additional proofs, our set of theorems is as follows:<br /><br /><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; text-align: center;"><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">{&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E0</i>&nbsp;,&nbsp;<i>E00</i></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">&nbsp;,&nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>E100</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><i>&nbsp;</i></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">,&nbsp;</span><i>E10&nbsp;</i>,&nbsp;<span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E000</i>, &nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E0100&nbsp;</i>, &nbsp;</span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><i>E1100&nbsp;</i>}</span></div></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><br /></span></div><div style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">Now, let's be more precise about the concept of proof in a formal system. A&nbsp;<b>formal proof</b>&nbsp;is a numbered sequence of theorems, each of which is justified by the use of an axiom or the application of an inference rule to one or more previous theorems in the sequence. Each step or theorem in the proof must be justified. The last line in the proof is the theorem that the proof establishes, while the previous lines are intermediate theorems, sometimes called <b>lemmas.&nbsp;</b></span><br /><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;"><br /></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px;">As an example, here is the proof of the theorem <i>E0100</i> in <i>FS</i>:&nbsp;</span></div><div><ol><li><i>E0</i> [by A1]</li><li><i>E00</i> [by IR1 applied to theorem on line 1]</li><li><i>E100</i>&nbsp;[by IR2 applied to theorem on line 2]</li><li><i>E0100</i>&nbsp;[by IR1 applied to theorem on line 3]</li></ol><div>In the sample proof given above, the justifications are given in square brackets.<br /><br />Now your turn. Here are a few exercises:<br /><ol><li>Is <i>E1010</i> a theorem of <i>FS</i>? Prove your answer.</li><li>Is <i>E0101</i> a theorem&nbsp;of&nbsp;<i>FS</i>? Prove your answer (Watch out: this one is tricky).</li><li>What is common to all of the theorems of <i>FS&nbsp;</i>?</li><li>What is common to all of the non-theorems of&nbsp;<i>FS</i>, that is, to all of the strings built upon the alphabet of <i>FS</i> that are not theorems of <i>FS&nbsp;</i>?</li><li>Can you find an interpretation or meaning for <i>FS</i>, that is, a way to interpret the theorems of <i>FS</i> as true sentences in a restricted domain and the non-theorems of <i>FS</i> as false or nonsensical sentences in the same domain?</li></ol><div>The main learning outcome for these exercises is for you to realize when you are working INSIDE the formal system and when you are working OUTSIDE the formal system.</div><div><br /></div><div>More on this in a later post.</div></div></div></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/r_0AsAJdOHU" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com0http://summerofgodel.blogspot.com/2013/05/formal-systems-axioms-inference-rules.htmltag:blogger.com,1999:blog-6792710671733445593.post-33857225879763290562013-05-15T17:54:00.000-07:002013-06-01T09:02:06.707-07:00Let's get started with Nagel and Newman (Section I)Let's ease our way into this. I haven't read the Nagel and Newman book in a few years. But I remember it as a short and highly readable overview of the proof. I have a 1986 softcover edition by NYP Press. Today, I want to write about Section I - Introduction. It's really short: under 5 pages.<br /><br />First, a short quote that makes me feel better about my past failures at REALLY understanding the proof of GIT:<br /><a name='more'></a><br /><blockquote class="tr_bq">"The details of Gödel's proof in his epoch-making paper are too difficult to follow without considerable mathematical training."&nbsp;(page 7)</blockquote>Nevertheless, the book aims to convey "the basic structure of his demonstrations" to "readers with very limited mathematical and logical preparation." Check!<br /><br />The bulk of the introduction discusses, in very general terms,&nbsp;deductive reasoning, axiomatic systems, and logical proofs. Since we'll have many occasions to go into these topics in great depth over the next few months, I won't go into it now. In fact, I'll devote my next post to formal systems.<br /><br />Going back to the introduction, it makes several strong claims that I want to investigate after I master the proof, namely: <br /><ol><li>GIT are "revolutionary in their broad philosophical import." I definitely want to read about the philosophy of mathematics. But aren't there big claims made about the impact of GIT on the philosophy of mind, artificial intelligence, etc.? Controversial stuff, no doubt!</li><li>GIT "undermined deeply rooted preconceptions and demolished ancient hopes that were being freshly nourished by research on the foundations of mathematics." I assume this is a reference to formalism and David Hilbert's program.</li><li>GIT "introduced into the study of foundation questions a new technique of analysis comparable in its nature and fertility with the algebraic method that René Descartes introduced into geometry." I wonder what "new technique" the authors are referring to here. Gödel numbering or some other aspect of the proof?</li></ol><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/kAHOPF48cg8" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com2http://summerofgodel.blogspot.com/2013/05/lets-get-started-with-nagel-and-newman.htmltag:blogger.com,1999:blog-6792710671733445593.post-7498171291646943722013-05-14T19:04:00.001-07:002013-06-01T09:00:47.106-07:00I just want to understand the proof of Gödel's incompleteness theoremsSo this is it! Summer 2013...&nbsp;This is when I get to master the proof of Gödel's incompleteness theorems (thereafter: GIT).<br /><br /><div>I discovered GIT as an undergraduate student in one of my Artificial Intelligence courses. My instructor was reading <a href="http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach"><span class="Apple-style-span" style="color: blue;">Gödel, Escher, Bach</span></a> and was raving about the book. He even took some of our exam problems from it. But what I remember most is the outline of the proof of GIT, more precisely the first theorem. To be honest, the only part of it that stuck with me was the idea of numbering the formulas. Of course, at the time, I bought a copy of the book. I liked many parts of it but I never finished it.</div><div><br /></div><div>Years later, while I was teaching computability theory and other computer science topics, I encountered GIT several times. At some point, I decided I wanted to understand the proof of GIT.&nbsp;</div><div><br /></div><div>First, I read the classi<span class="Apple-style-span" style="background-color: white;">c</span> <a href="http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371" style="color: blue;">"Gödel's Proof" by Nagel and Newman</a>. That is a concise and user-friendly overview of the proof. But it was only a proof sketch. So I looked at some textbooks on formal logic, which were neither concise nor user-friendly. I still did not understand the proof.</div><div><br /></div><div>Finally, a couple of years ago, I came across a promising volume: <a href="http://www.logicmatters.net/igt/"><span class="Apple-style-span" style="background-color: white; color: blue;">"Introduction to Gödel’s Theorems" by Peter Smith</span></a>. I read the first few chapters but then ran out of time (and motivation) to finish it. The following year, I made another attempt: I had to start from scratch and did not get any farther into it the second time around...<br /><br /><a name='more'></a><br />So what is it going to take for me to understand the proof of GIT?&nbsp;Maybe a blog... I know that sounds weird. So I'll try to explain.<br /><br />Now that I have a nearly-four-month-long summer ahead of me with no teaching duties ("only" research projects will occupy my mind and my time), I have decided to give it another try. I am going to use this blog to take notes on what I am reading. I will write up the new stuff I just learned. I will write down unanswered questions I still have to explore, etc. Most importantly, I will force myself to post (at least) one entry each week. If I have made no progress during the week, I will have to admit it here, in writing. Will that be enough motivation to keep me going until I reach my goal? We'll see...<br /><br />And what if nobody reads these posts? Ha! See, that cannot happen, because *I* am the primary target audience for this blog! And I know I will (re)read these posts.<br /><br />Nevertheless, if you stumble upon this blog and would like to comment, fill in some blanks in my understanding, add reading suggestions, argue with my interpretations, etc., please be my guest. I would love to share my passion for GIT with fellow enthusiasts.<br /><br />But for now, I guess it'll just be me and my buddy Kurt!<br /><br /><br /><br /><br /></div><img src="http://feeds.feedburner.com/~r/SummerOfGodel/~4/NaAbsyKOt9Q" height="1" width="1" alt=""/>David Furcyhttp://www.blogger.com/profile/12284114397975852146noreply@blogger.com2http://summerofgodel.blogspot.com/2013/05/i-just-want-to-understand-proof-of.html